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Any replacements are listed farther down
[3647] viXra:2603.0035 [pdf] submitted on 2026-03-06 21:25:09
Authors: René-Louis Clerc
Comments: 15 Pages. In French
Prime numbers are essential in various areas of cybersecurity, from encrypted messages to secure payments, HTTPS websites, and RSA cryptography techniques.Numerous computational and theoretical studies on these "unbreakable" numbers in arithmetic ((+)) are regularly published. Here we continue the article [15] by considering sequences of consecutive primes of the type considered then, primes with a certain initial digit, primes without a certain digit, primes with a certain digit, primes with a certain last digit, primes with a certain digit in a certain position, etc. We will calculate the maximum number of consecutive elements in such sequences, as well as the probabilities of obtaining a certain consecutive number from one of these primes. We will also seek to determine whether there is a possible order for the maximum lengths of sequences of such primes with reference to these various digits, and in which cases this may be a Benford-type order (inverse natural order of integers). We will conclude with some results concerning the maximum differences between any consecutive primes or those possessing a certain property.
Category: Number Theory
[3646] viXra:2603.0023 [pdf] submitted on 2026-03-04 21:29:50
Authors: Michael E. Spencer
Comments: 99 Pages.
This work develops a refinement—deterministic arithmetic framework for the odd-to-odd Collatz dynamics. The admissible inverse mapR(n; k) = (2^k n - 1) / 3is governed locally by residue—phase conditions on the live classes 1 and 5 (mod 6) and refines coherently through the exponential modulus towerM_j = 2 * 3^(j+1).At each level, admissibility of finite k-words depends only on the residue modulo M_j, and refinement introduces additional phase coordinates without ambiguity.Globally, admissible inverse lifts generate disjoint affine rails whose minimal bases are uniquely determined. Independently, the dyadic valuationk = v_2(3m + 1)produces an exact slice decomposition of the odd integers with weights 2^(-k). We prove that the affine rail partition and the dyadic slice decomposition coincide exactly, yielding a single unified arithmetic structure in which every odd integer possesses a unique admissible ancestry.A refinement-induced acyclicity principle is established: no finite admissible k-word remains compatible across all refinement levels M_j. Periodic inverse instruction regimes are destroyed by phase shifts under refinement, excluding nontrivial odd cycles. Moreover, compatibility across the refinement tower forces every infinite admissible chain to realize a base residue in the anchor structure; hence no divergent trajectory can occur.Finally, the forward mapT(m) = (3m + 1) / 2^(v_2(3m + 1))is shown to be the exact algebraic inverse of all admissible inverse lifts. Forward and inverse dynamics therefore coincide on a single closed affine system anchored at 1.Consequently, the odd-to-odd Collatz dynamics admit a complete internal arithmetic classification, and every forward trajectory converges to the fixed point 1.
Category: Number Theory
[3645] viXra:2603.0021 [pdf] submitted on 2026-03-04 12:08:28
Authors: Óscar E. Chamizo Sánchez
Comments: 4 Pages.
The twin prime conjecture, asserting there are infinitely many pairs of primes differing by 2, was popularized by French mathematician Alphonse de Polignac in 1849. We are pleased to present an astounding and overwhelming proof with a clasic "reductio ad absurdum" flavour revealing, by the way, a perhaps not so amazing relationship with the Goldbach conjecture and testing, since the core of reasoning is the same, that both statements are strongly connected.
Category: Number Theory
[3644] viXra:2603.0014 [pdf] submitted on 2026-03-03 21:33:33
Authors: Xian Wang, Luoyi Fu
Comments: 55 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by extending the Riemann zeta function and Dirichlet $L$ -functions to the elliptic complex domain, based on a newly constructed system of elliptic complex numbers $mathbb{C}_lambda(lambda<0)$ . The core challenge addressed is the inherent difficulty in resolving these conjectures within the traditional "circular complex domain" framework ($lambda=-1$); the author posits that a complete proof is unattainable strictly within this conventional setting.The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers, specifically identifying the limiting case as $lambdato 0^{-}$ as the key to the proof. Through rigorous deduction, a bijective correspondence between zeros across different complex planes is established. By employing proof by contradiction and leveraging the correspondence between $mathbb{C}_lambda$(as $lambdato 0$) and the circle complex plane $mathbb{C}$, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven.This paper is organized into three parts:begin{enumerate}item Construction and Geometric Properties: The first part details the construction of elliptic complex numbers and their fundamental geometric properties, laying the necessary foundation for subsequent analysis and the proof of the conjectures.item Analytic Extension: The second part introduces elliptic complex numbers into mathematical analysis, deriving numerous results analogous to those in classical complex variable function theory.item Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and the Generalized Riemann Hypothesis.
Category: Number Theory
[3643] viXra:2603.0013 [pdf] submitted on 2026-03-03 21:28:10
Authors: Christoper Mututu
Comments: 36 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We investigate a previously undocumented integer transformation whose explosive behavior places it beyond the known extremes of Collatz-type dynamics. The system operates by (1) squaring each digit of an integer and concatenating the results, then (2) repeatedly compressing the expanded sequence by summing every consecutive triplet of digits whenever the total length remains divisible by three. This deceptively elementary process generates combinatorial shockwaves: numerical structures routinely balloon to hundreds or thousands of digits before undergoing catastrophic collapse into a microscopic attractor set. [etc.]
Category: Number Theory
[3642] viXra:2603.0009 [pdf] submitted on 2026-03-01 22:19:46
Authors: Edward C. Larson
Comments: 5 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)
A novel derivation of the density of the distribution of prime numbers is presented, based on a simple frequentist analysis and the smallest scale at which a rigorous upper bound on the frequency holds. An approximating differential equation is derived. It is shown that in the asymptotic limit, the density of primes, pi(x), scales as x / ln x, in accordance with the Prime Number Theorem (PNT). The approach bridges the gap between discrete number theory and continuous differential modeling, offering a mechanistic explanation for the observed thinning of prime density that mirrors the foundational results of classical analysis.
Category: Number Theory
[3641] viXra:2602.0145 [pdf] submitted on 2026-02-24 21:55:54
Authors: Niccan Mandal
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, we derive a formal inversion identity from the Taylor expansion of $sqrt[k]{x}$ to get $x$ as an infinite series of the function. Along with the derivation, we also give a proof of the identity by justifying some crucial mathematically rigorous statements regarding analyticity, validity of Cauchy's convolution, and the convergence, and also derive a trivial infinite series for $pi$, $e$ (Euler's constant) and a formal infinite series identity of $gamma$ (Euler-Mascheroni constant) in terms of their $k$-th roots.
Category: Number Theory
[3640] viXra:2602.0144 [pdf] submitted on 2026-02-24 21:52:03
Authors: Marcin Barylski
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
There are several interesting properties of triangular numbers and research work devoted to them. One of the them is correlation between them and primes- there is hypothesis that between every two different triangular numbers >1 there is always a prime number. This paper is focused on detailed examination of difference, mainly between triangular numbers and their closest (smaller or greater) primes (this difference is called in this work delta, δTP), including its extreme values, also in spirit of finding effective test to search for prime numbers.
Category: Number Theory
[3639] viXra:2602.0139 [pdf] submitted on 2026-02-23 19:56:27
Authors: P. Murugesha
Comments: 2 Pages.
In this paper i going to compare matrix determinants and Beals conjuncture.. In this paper I consider equation (az^2-by^2)x^2-(bx^2-cz^2)y^2=(ax^2-cy^2)z^2.Next taken bracket containing things in above equation as x, y, z and proved equation 5 as no solution.... So,I concluded in this paper( if we replace (az^l-by^m)x^n-(bx^n-cz^l)y^m=(ax^n-cy^m)z^l then again taken this equation az^l-by^m=x and bx^n-cz^l=y and ax^n-cy^m=z(note l, m, n are >=2). Then also get no solution (it get same as equation 5 only changes in x, y, z powers) it still holds the properties of relatively prime properties.) So we finally got answer for beal conjuncture. When powers greater than 2 has no solution.
Category: Number Theory
[3638] viXra:2602.0131 [pdf] submitted on 2026-02-22 10:40:17
Authors: Timothy Jones
Comments: 2 Pages.
We use a TI-84 program to show pi's rationality implies that a radius at 90 degrees must have a defined slope, a contradiction.
Category: Number Theory
[3637] viXra:2602.0109 [pdf] submitted on 2026-02-21 19:37:24
Authors: Christoper Muoki Mututu
Comments: 13 Pages. (Note by viXra Admin: For the last time, Please cite and list scientific references!)
We investigate a structural subclassification of twin prime pairs based on intersections between two modular quadruplet configurations, an admissible (2, 4, 2) prime pattern and a complimentary forbidden quadruplet pattern eliminated modulo 3. We define an overlap counting function ��(��) measuring the number of twin primes up to �� arising from such structural intersections and compare it to the total twin prime count ��(��). Computational data up to ��=3×1011 shows that the ratio ��(��)=��(��) ��(��) increases from approximately 0.4 at 103 to approximately 0.6568 at 3×1011. We prove that the structural configurations underlying the overlap occur infinitely often as arithmetic patterns and that ��(��)→∞ ���� ��→∞. We do not prove infinitude of twin primes nor do we establish a limiting value of ��(��). However, the data suggests that the overlap subclass forms a substantial and stable proportion of observed twin primes at large computational scales. This work provides an empirical decomposition of twin primes that they may compliment probabilistic models such as the Hardy-Littlewood heuristic.
Category: Number Theory
[3636] viXra:2602.0108 [pdf] submitted on 2026-02-20 20:35:02
Authors: Arthur V. Shevenyonov
Comments: 4 Pages.
An introduction to L-gebra, a promising algebraic apparatus spanning areas as diverse as calculus & number theory to name but a few, and bridging the otherwise distinct if disparate operations & operators, should suffice for a potent yet succinct treatment of Fermat’s augmented last proposition & Riemann’s hypothesis.
Category: Number Theory
[3635] viXra:2602.0103 [pdf] submitted on 2026-02-19 20:16:54
Authors: Theophilus Agama
Comments: 8 Pages.
Denote the minimal length of a fixed degree d>1 addition chain that leads to n by l^d(n). We introduce the concept of a strong Brauer number of rank d>1 and show that all numbers belonging to this class satisfy the inequality l^d(d^n-1)
Category: Number Theory
[3634] viXra:2602.0095 [pdf] submitted on 2026-02-18 20:40:02
Authors: Francesco Aquilante
Comments: 6 Pages.
Beal Conjecture, which asserts that for $a^k + b^m = c^n$ with $k, m, n > 2$, the bases $a, b,$ and $c$ must share a common prime factor. We prove it to be true with an approach that utilizes a sequence of rational perturbations $delta={delta_i}:{delta}_{i in mathbb{N}} subset mathbb{Q}$ , $delta_i > 0$ and $lim_{i to infty} delta_i = 0$ to treat such Diophantine equation as the critical limit-state of a geometrically constrained configuration. By defining a sequence of non-degenerate triangles $mathcal{T}_delta$ with rational side lengths ${a^k, b^m, c^n - delta_i}$, we establish a continuous mapping to the moduli stack of elliptic curves $mathcal{M}_{1,1}$.We demonstrate that the requirement for {rationality of the configuration} (the existence of a rational altitude $h_delta$) induces a sequence of Frey-Hellegouarch curves $E_delta$ that converge algebraically to the limit-state $E_{Beal}$. For signatures where $min(k,m,n) geq 3$, we invoke Ribet’s Level-Lowering Theorem to show that the associated Galois representation $ho_{E,n}$ is necessitated to reside within the {em empty space} of weight-2 cuspidal modular forms $S_2(Gamma_0(2))$.Simply speaking, our proof follows the often anticipated path of reasoning by which if Beal Conjecture were trueit must ultimately stand on the foundationthat underpins the validity of Fermat's Last Theorem.Furthermore, we provide a formal textit{Parity Lemma} to delineate the bifurcation at $n=2$, explaining why the modular sieve permits coprime solutions in Fermat-Catalan and Pythagorean signatures. This topological and arithmetic framework confirms that for strictly hyperbolic signatures, a solution exists if and only if $gcd(a, b, c) > 1$.
Category: Number Theory
[3633] viXra:2602.0090 [pdf] submitted on 2026-02-18 13:18:57
Authors: Óscar E. Chamizo Sánchez
Comments: 6 Pages.
An ancient conjecture, named after its discoverer as Goldbach conjecture [1][2], that is to say, every even number greater than 2 can be represented by the sum of two primes, is a simple and intractable statement that has been torturing mathematicians for more than 250 years. We wonder if the divide et impera method, so useful in programming and algorithmics, could provide some service here. The goal is simplify and separate the whole problem into three independent and fairly manegeable subproblems. An approach that, as far as I know, has not been tested before
Category: Number Theory
[3632] viXra:2602.0086 [pdf] submitted on 2026-02-17 00:46:47
Authors: Ryujin Choi
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
We study the distribution of integers obtained by removing fixed residue classes moduloprimes. Using an explicit upper-bound sieve argument, we show that admissible integers cannotoccupy arbitrarily long contiguous intervals. In the case of two arithmetic progressions, thisleads to the existence of simultaneous prime values. As a consequence, Goldbach’s conjectureand the twin prime conjecture follow.
Category: Number Theory
[3631] viXra:2602.0073 [pdf] submitted on 2026-02-12 19:56:03
Authors: Francesco Aquilante
Comments: 7 Pages.
We present the quantum Riemann sum ($Q$-sum) operator framework and use it to prove the irrationality of the Riemann-$zeta$ function at odd integers, the Dirichlet-$beta$ function atat all positive integers $n geq 2$, as well as that of the Euler-Mascheroni constant ($gamma$). By establishing a recursive functional hierarchy, we circumvent the classical ``parity barrier'' that has traditionally isolated even and odd zeta-type constants. We utilize the $p$-adic Newton Polygon to demonstrate that the arithmetic complexity of the operator kernel is an invariant of the functional hierarchy. Therefore, the irrationality of the transcendental anchors $zeta(2)$ and $beta(1)$ necessitates the irrationality of the entire chain. This line of reasoning can be extended to incorporate $gamma$, thereby substantiating its long-held irrationality.
Category: Number Theory
[3630] viXra:2602.0063 [pdf] submitted on 2026-02-09 18:15:31
Authors: Edgar Valdebenito
Comments: 9 Pages.
In this note, we study the sum S=(1/2)+(1/10)+(1/30)+(1/68)+...
Category: Number Theory
[3629] viXra:2602.0054 [pdf] submitted on 2026-02-06 21:13:45
Authors: Ammar Hamdous
Comments: 18 Pages. Creative Commons Attribution 4.0 International
In earlier work [1], we introduced a refined and more structurally representative Collatz tree, within which we identified a singularity. A subsequent preprint [2] established a methodological generalization of the Collatz sequences that preserves this singularity by extending it to a generalized singularity. In the present paper, we investigate the structure of the generalized Collatz tree—referred to as the k-Tree—arising from this transformation. Our analysis focuses on the ordering, propagation, and interaction of branch beginnings across ranks, with particular attention to the structural sets Bk and Ak. This study aims to elucidate the internal architecture of the generalized tree and to clarify the extent to which the geometric and dynamical features of the classical Collatz tree persist under the generalization.
Category: Number Theory
[3628] viXra:2602.0036 [pdf] submitted on 2026-02-06 19:36:32
Authors: Silvana di Savino
Comments: 5 Pages.
The even number 2n, which is the product of two or more prime numbers with 2, is always equal to the sum of only two prime numbers equidistant from half their sum; the odd number 2n+1, which is the product of two or more prime numbers with 2 + the odd number 1, is always equal to the sum of two prime numbers equidistant from half their sum, which is an even number + a prime number, 1+2n, which is the difference between the two equidistant primes.
Category: Number Theory
[3627] viXra:2602.0031 [pdf] submitted on 2026-02-05 20:18:03
Authors: Theophilus Agama
Comments: 19 Pages.
We denote the length of an addition chain with fixed degree d>2 leading to n by l^d(n). We study the counting function F_d(m,r):=#{nin [d^m,d^{m+1})~:~ l^d(n)<m+r} establishing upper and lower bounds, which generalizes previous classical investigations of De Koninck, Doyon, and Verreault.
Category: Number Theory
[3626] viXra:2602.0024 [pdf] submitted on 2026-02-04 23:16:46
Authors: Sriramadesikan Jagannathan
Comments: 11 Pages. (Note by viXra Admin: Please cite and list scientific references)
This paper presents a proof of the Riemann Hypothesis by examining the geometric and arithmetic properties of the Dirichlet eta function. By assuming the existence of zeros off the critical line, and analyzing the resulting alternating series in the complex plane, we establish a logical contradiction. The proof relies on insights into the structure of these series, demonstrating that all non-trivial zeros must possess a real part of exactly 1/2.
Category: Number Theory
[3625] viXra:2602.0018 [pdf] submitted on 2026-02-03 20:26:41
Authors: Srihan Dutta, Subhraneel Dutta
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper investigates two specific modular exponentiation identities involving fixed integers. First, we determine the set of non-negative integers m satisfying aN ≡ am (mod N) for a fixed N > 1 and all integers a, deriving the minimum such m. Second, we analyze the minimum positive integer n such that amn ≡ an (mod x) holds for a fixed x > 1 and all integers a, m. We provide explicit formulas for these minimal exponents in terms of the prime factorization exponents and the Carmichael function λ(·).
Category: Number Theory
[3624] viXra:2602.0008 [pdf] submitted on 2026-02-01 19:50:12
Authors: Steven M. Tylock
Comments: 8 Pages.
The Collatz conjecture offers a seemingly arbitrary piecewise sequencing of two separate functions (divide by two, multiply by three and add one). Attempts have been made to partially simplify the problem by combining exactly one instance of the multiplication with one instance of the division but have not previously been able to completely separate the two alternatives. To create this separation, I define a positive integer’s Least Significant Bit as the smallest power of two that is added together to create its binary representation. I then define a replacement function as three times n plus the Least Significant Bit of n. I then show that an application of the replacement function followed by division by two has an identical result to division by two followed by the original Collatz multiplication. By using the replacement function, all division can be delayed until the result is a perfect power of two. This change removes the piecewise aspect of the Collatz conjecture that has stymied a proof. In addition, the resulting graph of transformations displays a many-to-one relationship that has previously been hidden. The replacement formula’s non-piecewise and many-to-one features offer new avenues to prove the conjecture. If one can prove that the replacement function reaches a perfect power of two, one will have proved the Collatz.
Category: Number Theory
[3623] viXra:2602.0007 [pdf] submitted on 2026-02-02 02:41:50
Authors: Jonipol E. Fortaliza
Comments: 18 Pages.
Throughout mathematics history, mathematicians had created triangular array of numbers. Famous among these number triangles is the Pascal’s Triangle which had marked its prominence in many areas of mathematics and even extends its usefulness in the sciences. This paper presents an inventory of number triangles known and recognized in the mathematics world and takes a look to newly-found triangular array of numbers generated by the function, and its link to the Pascal’s Triangle particularly to the Tetrahedral Numbers.
Category: Number Theory
[3622] viXra:2602.0005 [pdf] submitted on 2026-02-01 01:40:16
Authors: Md. Razib Talukder
Comments: 12 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
At a specific angle, the fundamental rule for triangles, the cosine rule simplifies to the same form as the equation in one of mathematics’ most famous problems. This connection arises when the angle is a right angle, linking a basic geometric idea with the only case for which the centuries-old statement holds true.
Category: Number Theory
[3621] viXra:2601.0140 [pdf] submitted on 2026-01-30 01:29:10
Authors: P. N. Seetharaman
Comments: 10 Pages.
This paper presents a straightforward and elementary proof of Fermat's Last Theorem (FLT), asserting that there are no integer solutions to a^n +b^n = c^n for n > 2. Leveraging basic number theory and algebraic manipulations, we offer a concise demonstration aiming to make this fundamental result accessible to a broad mathematical audience.
Category: Number Theory
[3620] viXra:2601.0133 [pdf] submitted on 2026-01-27 01:58:49
Authors: Walter A. Kehowski
Comments: 9 Pages.
A power spectral number is a positive integer whose spectral basis consists of only primes and powers. If one searches for power spectral numbers whose spectral sum is also a power, then one finds only five examples. We call these numbers power spectral Pythagorean numbers. The first two examples involve the Pythagorean triples 3,4,5 and 8,15,17. It is shown in this note that these are the only two Pythagorean triples that are power spectral Pythagorean. The other three examples involve the Pell equation.
Category: Number Theory
[3619] viXra:2601.0109 [pdf] submitted on 2026-01-23 08:53:31
Authors: Timothy Jones
Comments: 1 Page.
Using tangent lines to the unit circle, we give an argument that shows pi is irrational.
Category: Number Theory
[3618] viXra:2601.0103 [pdf] submitted on 2026-01-22 10:15:17
Authors: Rolando Zucchini
Comments: 17 Pages.
Since ancient Greece the possibility of defining natural numbers was considered, but, unlike what happened in Geometry in the Euclid’s Elements, all efforts were in vain. After 2000 years it was the Italian mathematician Giuseppe Peano who was recognized for the historical merit of having provided a rigorous definition of the natural numbers and their properties. His five postulates represent the first well-defined axiomatic foundation of arithmetic. Peano's fifth postulate, known as the Principle of Induction, has provided an indispensable tool in countless mathematical proofs and has enabled significant progress in understanding numbers and their secrets. This paper contains numerous solved exercises on the application of Induction Principle.
Category: Number Theory
[3617] viXra:2601.0097 [pdf] submitted on 2026-01-22 21:24:24
Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages.
In this paper, assuming that the conjecture c [smaller than] R*2 is true, we give the proof that the explicit abc conjecture of Alan Baker is true and it implies that the abc conjecture is true. We propose the mathematical expression of the constant K(epsilon). Some numerical examples are provided.
Category: Number Theory
[3616] viXra:2601.0094 [pdf] submitted on 2026-01-22 21:11:17
Authors: T. Nakashima
Comments: 4 Pages.
Riemann Hypothesis has been the unsolved conjecture for 164 years. This conjecture is the last one of conjectures without proof in "{U}eber die Anzahl der Primzahlen unter einer gegebenen Gr"{o}sse"(B.Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2. Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper,I guess the independence (unprovability) of the Riemann Hypothesis. I consider the axiomatic system in which the Riemann Hypothesis holds and the axiomatic system in which the Riemann Hypothesis does not hold. Finally, I guess that the Riemann Hypothesis is unprovable.
Category: Number Theory
[3615] viXra:2601.0089 [pdf] submitted on 2026-01-21 06:33:14
Authors: Ryan Hackbarth
Comments: 8 Pages.
Here I present a derivation of an equation whose solution sets are the trivial and nontrivial zeros of the Riemann Zeta Function. I demonstrate how the trivial solutions are directly encoded by integer inputs and how these can be mapped by a symmetry to positive odd integers. I extend this insight to encode the even integers, and map these to the negative odd integers, which provides an explicit connection between particular values of the Riemann Zeta Function which have historical and ongoing research interest. I then extend this symmetry to the nontrivial zeroes, and demonstrate the dependence of the critical line in producing this symmetry. Finally, I note the distribution of the nontrivial zeroes have a correspondence with the distribution of trivial zeroes, and provide a first order approximation of this correspondence.
Category: Number Theory
[3614] viXra:2601.0068 [pdf] submitted on 2026-01-16 21:32:20
Authors: Ryan Hackbarth
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references!)
Here I present an equation for the Zeros of the Riemann Zeta Function which connects the distribution of the trivial zeroes with integer inputs to the distribution of the nontrivial zeroes. I demonstrate that this relationship explicitly depends on the critical line where a = ½. I do so in plain language and with replicable calculations, as when I try to write like a mathematician it comes across as inauthentic and bad. Finally, I provide an appendix of calculated solutions.
Category: Number Theory
[3613] viXra:2601.0067 [pdf] submitted on 2026-01-16 03:22:57
Authors: Juan Francisco Petitti
Comments: 30 Pages. (Note by viXra Admin: Author is required in the article after article title)
This paper presents a definitive proof of the Riemann Hypothesis by establishing the existence of a self-adjoint operator whose spectrum corresponds precisely to the non-trivial zeros of the Riemann zeta function. Utilizing the framework of quantum mechanics and the strong convergence of a sequence of operators Hn in the supremum norm, we demonstrate that the eigenvalues of the limiting operator are real. We show that the Riemann zeta function is an entire function of order one, satisfying the conditions of the Hilbert-Pólya conjecture. The results confirm that all non-trivial zeros lie on the critical line Re(s)=1/2, thereby resolving the most significant open problem in number theory.
Category: Number Theory
[3612] viXra:2601.0066 [pdf] submitted on 2026-01-15 06:55:09
Authors: J. Kuzmanis
Comments: 9 Pages.
A mathematically simple odd semiprime factorization method is presented.
Category: Number Theory
[3611] viXra:2601.0059 [pdf] submitted on 2026-01-14 17:01:00
Authors: Dmitriy S. Tipikin
Comments: 3 Pages.
A famous Fibonacci sequence is forming a simple cycle when sign plus is replaced to minus. A simple proof for any numbers is outlined.
Category: Number Theory
[3610] viXra:2601.0052 [pdf] submitted on 2026-01-13 22:50:17
Authors: Youssef Ayyad
Comments: 24 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Prime numbers have traditionally been studied through the austere lens of arithmetic, yet their deepest structure may be geometric in nature. This work presents a paradigm shift: we construct a toroidal manifold (mathbb{T}^2) where integers are mapped via the phase embedding (Phi(n) = sqrt{n} e^{isqrt{npi}}), transforming discrete divisibility into continuous phase orthogonality. The geometric dust—the area remainder (R(n) = pi n^2 - frac{1}{2}n^3sin(2pi/n))—accumulates into a quantum Hamiltonian (H = -Delta + V) on (mathbb{T}^2). We prove (H) is self-adjoint and its spectrum ({lambda_j}) exhibits Gaussian Unitary Ensemble (GUE) statistics, as verified numerically. Crucially, we propose a textbf{geometric formulation} of the Riemann Hypothesis: we show that, under the assumption of RH, the eigenvalues of (H) are real, bounded below by (frac14), and satisfy the spectral correspondence (lambda_j^{text{(calibrated)}} = frac14 + t_j^2), where (frac12 + it_j) are the non-trivial zeros of (zeta(s)). Numerical verification shows agreement within (0.1%) for the first 50 zeros. The framework reveals primes as ground-state singularities in a resonant field, offering an intuitive geometric foundation for their distribution—not as a proof of RH, but as a novel geometric-spectral formulation of it. For recent developments in geometric approaches to number theory, see Kontorovich and Nakamura (2022), Sarnak (2021), and the survey by Baluyot (2023) on spectral approaches to zeta zeros.
Category: Number Theory
[3609] viXra:2601.0050 [pdf] submitted on 2026-01-13 01:28:26
Authors: Debasis Biswas
Comments: 03 Pages.
In this paper Polya equivalent of Riemann Hypothesis is proved from Complex analytic expression of Riemann Xi function.
Category: Number Theory
[3608] viXra:2601.0046 [pdf] submitted on 2026-01-12 20:50:18
Authors: Youssouf Ouédraogo
Comments: 21 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org) Creative Commons Attribution 4.0 International
This paper proposes a new structural approach to the study of consecutive prime numbers based on a quadratic relation linking three successive primes. A stability ratio is introduced and shown to converge asymptotically to unity using explicit bounds for the k-th prime number. This convergence induces a constraint on the local variation of prime gaps, leading to an asymptotic smoothness law for their relative fluctuations. The analysis is fully deterministic and avoids heuristic arguments based on average asymptotic. Numerical validations using verified large prime datasets confirm the theoretical predictions and illustrate the progressive regularization of local gap variations as the prime index increases.
Category: Number Theory
[3607] viXra:2601.0039 [pdf] submitted on 2026-01-11 01:30:39
Authors: Ryan Hackbarth
Comments: 2 Pages.
In this paper, I present a formula for the zeroes of the Riemann Zeta Function and highlight their dependence on a rational integer ratio. I connect these ratios with a hyperbola reminiscent of Pell’s equation which approximates pi and provide a table of calculated ratios and their corresponding Zero. Finally, I demonstrate the requirement of the critical line at ½ in producing these integer approximations.
Category: Number Theory
[3606] viXra:2601.0024 [pdf] submitted on 2026-01-06 14:29:32
Authors: Brian Scannell
Comments: 25 Pages.
To support intuitive understanding of Fermat’s Last Theorem, this paper presents a simple visualisation based on a defined normalised Fermat plot and shows that rational directions arising from succession t Pythagorean triples—with a fixed hypotenuse gap—become automatically irrational beyond a finite point, explaining why no Fermat type integer solutions can occur along these directions.
Category: Number Theory
[3605] viXra:2601.0020 [pdf] submitted on 2026-01-05 20:31:18
Authors: Silvio Gabbianelli
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references)
By observing the relative positions of odd composite numbers in the set of odd natural numbers up to a given n, the positions of the prime numbers can be logically derived by subtraction. Not only that, but a linear, albeit parametric, function can also be deduced that can provide all and only the odd composite natural numbers up to n, and therefore all the prime numbers up to n. This allows us toformulate the conjecture that the set of prime numbers (except 2) is the well-ordered complementary set of odd composite numbers. This ordering can also be seen using the Cartesian line y = 2x + 1. Other lines and different numberings can highlight other possible properties of prime numbers.
Category: Number Theory
[3604] viXra:2601.0016 [pdf] submitted on 2026-01-04 14:36:58
Authors: Giovanni Di Savino
Comments: 5 Pages.
Thales measured the height of the inaccessible pyramid and the distance of the unreachable ship from the harbor, demonstrating that anything that can be plotted on a plane can be measured; Euclid, with the product of known prime numbers, continually generates new primes and demonstrated that prime numbers are infinite; Peano, with the second of his five axioms, affirmed that for every natural number there exists a successor number +1. We will never be able to claim to have developed Euclid's inaccessible primes or Peano's unattainable number, but twin primes are two of the infinite primes, one of which is a successor number +2 of the other prime, and the sum of the two primes is always a number 6n; by representing even numbers in the form 6n or 6n±2 and odd numbers in the form 6n±1 or 6n±2±1, we can demonstrate that Euclid's inaccessible primes and Peano's unattainable successor number exist. All prime numbers, all twin primes, all Mersenne primes which are the sum of numbers in double proportion and generate the even perfect numbers, all odd numbers 3n of the Collatz algorithm whose successor +1 is a power 2^n_even which when halved is 2^(n-1) and ends at 2^0 = 1 and all even numbers and all odd numbers which are the sum of 2 or 3 primes, all exist and, even if they will never be known, the final digit of the prime numbers and of the successor number which can be a prime or composite number is known.
Category: Number Theory
[3603] viXra:2601.0001 [pdf] submitted on 2026-01-01 04:30:08
Authors: Keshava Prasad Halemane
Comments: 14 pages 1 table
The convergence of the Collatz-Hasse-Syracuse-Ulam-Kakutani Sequence is proved, thus proving the Collatz Conjecture, which has been an unsolved problem. The proof is based on the isomorphism established between the set of positive integers and a carefully designed system with a hierarchy (arborescence) of binary exponential ladders defined on the set of positive odd numbers.
Category: Number Theory
[3602] viXra:2512.0128 [pdf] submitted on 2025-12-27 01:18:45
Authors: Arthur V. Shevenyonov
Comments: 4 Pages.
A simple-yet-fulfilling formula is presented showcasing a linkage between the prime number’s[shifted] index (ordinal rank) vs its signature trace, or power sum over the characteristic 2-basis. A loose analogy of minimum action or energy orbitals could be conceived as one way ofrationalizing the representation from amongst alternate, likewise 2-basis, candidates. A mostminimalist calculus is proposed accommodating such singular-basis travel.
Category: Number Theory
[3601] viXra:2512.0111 [pdf] submitted on 2025-12-24 01:09:27
Authors: Theophilus Agama
Comments: 6 Pages.
We extend the inequality due to Alfred Brauer on standard addition chains to a sequence of additions leading to a finite number where at most at most $dgeq 2$ previous terms can be added to generate each term in the sequence.
Category: Number Theory
[3600] viXra:2512.0110 [pdf] submitted on 2025-12-24 01:07:10
Authors: Arthur Shevenyonov
Comments: 3 Pages.
This paper proposes a ‘naïve,’ ‘three-line’ demonstration for RH building on a functional-equation reduction. It is suggested, inter alia, how prematurely restricting an exposition to the conjectured critical band a priori may usher in some paradoxic singularities that unduly restrict the complex candidate (imaginary extension) domain, albeit without questioning the real core. It remains to be judged whether the ‘gray area,’ as implied or straddled, qualifies as a ‘constructive’ completion of the RH around its frontier of inference.
Category: Number Theory
[3599] viXra:2512.0109 [pdf] submitted on 2025-12-24 01:08:41
Authors: Arthur Shevenyonov
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific references)
A ‘naïve’ look into simple & augmented natural power-difference forms (nPDF) unleashes implications & patterns in areas as diverse as, the RH, primality formulae, and structure parallels, to name but a few.
Category: Number Theory
[3598] viXra:2512.0090 [pdf] submitted on 2025-12-19 21:58:53
Authors: Farhad Aliabdali
Comments: 7 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
This paper presents a complete, closed-form mathematical equation that exactly computes the prime-counting function π(N) for any integer N ≥ 2. Unlike existing methods which are either asymptotic approximations or recursive algorithms, our formulation is a single evaluable expression. The equation operates in two distinct modes: (1) using a sequence of known primes, or (2) using the simple sequence Ju2081 = 2, Ju2099 = (n-1)-th odd integer ≥ 3 for n ≥ 2, with an intrinsic primality test μu2099 = ⌈∏u2093u208cu2081u207fu207b¹ (Ju2099/Ju2093 - ⌊Ju2099/Ju2093⌋)⌉ where μu2099 = 1 if and only if Ju2099 is prime. The formula directly yields π(N) through elementary arithmetic operations without recursion, iteration, or algorithmic procedures. The implications of this formula are explored in comparison to existing prime counting functions and its potential impact on the study of prime distribution, it is an explicit sieve-theoretic expression and a self-contained rewriting. This complements classical exact prime-counting methods (Meissel—Lehmer and descendants), which are vastly more efficient for computation.
Category: Number Theory
[3597] viXra:2512.0089 [pdf] submitted on 2025-12-19 21:59:09
Authors: Umberto Bartocci, Alessandro Miotto
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
Given any natural number n, we consider the subset C(n) of all natural numbers which could be written in decimal basis as a string of length n (n-numbers). We are looking for subsets of C(n) which appear interesting in connection with n-prime numbers and twin n-prime numbers too. Those numbers consisting of only the digits 1 and 7 (that we call Sofia numbers) appear to be quite promising in this context, and their study suggests some natural conjecture.
Category: Number Theory
[3596] viXra:2512.0079 [pdf] submitted on 2025-12-17 06:26:11
Authors: Minho Baek
Comments: 2 Pages.
The purpose of this paper is to introduce regularity on Mersenne number with n=odd number. There is regularity among n=odd numbers in Mersen number. If the k and l is odd number, 2^kl-1=α(2^k-1).
Category: Number Theory
[3595] viXra:2512.0072 [pdf] submitted on 2025-12-16 03:47:28
Authors: Dominique Tremblay
Comments: 60 Pages.
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of ten postulates concerning twin primes, as well as a sample of nine additional hypotheses of the order of conjecture.
Category: Number Theory
[3594] viXra:2512.0070 [pdf] submitted on 2025-12-16 03:29:37
Authors: Farhad Aliabdali
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
The Collatz map T(n)=n/2 for even n and T(n)=3n+1 for odd n admits classical affine descriptions via parity vectors, but these typically compress each odd event into the macro-step (3n+1)/2, obscuring intermediate algebraic states. We introduce a two-stage expansion that separates an odd event into a rewrite step R (expressing n=2x+1) followed by a forced follow-up C (sending x↦3x+2), alongside the even halving step E. This yields a word system over {E,R,C} and a uniform normal formX_N (w)=(3^k(w) X_0+2^D(w) -3^k(w) +σ_N (w))/2^D(w) ,where σ_N (w) admits an explicit signed monomial expansion in powers of 3 and 2. We prove that complete two-stage words (those with every R immediately followed by C) compress under RC↦O to the standard parity-vector affine form, giving a precise equivalence criterion and a canonical matching rule (k,D,Σ). Consequently, removing the standard-image equations from the two-stage enumeration leaves exactly the truncated (dangling-R) equations corresponding to intermediate states not representable in the standard form. Finally, we derive residue-class "locking" conditions modulo 2^D(w) , clarifying integrality constraints and connecting the framework naturally to 2-adic formulations.
Category: Number Theory
[3593] viXra:2512.0065 [pdf] submitted on 2025-12-16 02:33:57
Authors: Zhi Li, Hua Li
Comments: 8 Pages.
Hyperreal numbers are a field encompassing real numbers, infinitesimals, and infinities. In thehyperreal number system, infinitesimals are not equal to zero, and operations can be performedon infinitesimals and infinities.Based on hyperreal number theory, this paper discovers that multi-level radicals that cannot besimplified have no definite value, varying in size; these can be called quantum numbers or inaccurate numbers. A special type of multi-level radical can exhibit quantum superposition phenomena similar to those in the physical world. This allows the hyperreal number system to be extended to a supernumber system, a field encompassing real numbers, infinitesimals, infinities,and inaccurate numbers.Confirming the existence and properties of quantum numbers or inaccurate numbers and classifying them as supernumbers provides a new perspective. Within this framework, the negation proves the Riemann hypothesis.
Category: Number Theory
[3592] viXra:2512.0064 [pdf] submitted on 2025-12-16 03:47:14
Authors: Dominique Tremblay
Comments: 56 Pages.
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of ten postulates concerning twin primes, as well as a sample of nine additional hypotheses of the order of conjecture.
Category: Number Theory
[3591] viXra:2512.0054 [pdf] submitted on 2025-12-12 00:41:14
Authors: Laurent Nedelec
Comments: 25 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This new text on the Collatz/Syracuse problem is a continuation of the document published in February 2025 on viXra (Laurent Nedelec : An algorithmic approach to solving the Collatz/Syracuse problem. viXra 2502.0056). It explains why the probabilities of divergence for Syracuse trajectories are extremely small. The same conclusion is obtained for non-trivial cycles: their existence is nearly impossible. These two results reinforce the conclusion of the previous text—reached through different methods—namely that the Collatz conjecture is, with very high probability, true.In this new work, we first analyze the structure of the alternations between even and odd iterations within Syracuse trajectories. We then show that N* has an equiprobable structure with respect to even and odd iterations in Syracuse trajectories. Next, we examine how this equiprobability within the structure of N* leads trajectories to be globally decreasing. Finally, we address the implications of these results for the questions of divergent trajectories and non-trivial cycles.
Category: Number Theory
[3590] viXra:2512.0052 [pdf] submitted on 2025-12-10 09:40:35
Authors: Miroslav Sukenik, Magdalena Sukenikova
Comments: 4 Pages.
The calculation of this constant requires advanced numerical methods such as interval mathematics, using specialized software for high accuracy and tight error limits. In this article, we present a simpler method for deriving the Viswanath’s constant.
Category: Number Theory
[3589] viXra:2512.0039 [pdf] submitted on 2025-12-08 05:43:00
Authors: Minho Baek
Comments: 4 Pages.
The purpose of this paper is to introduce patterns in prime numbers and then primality test from those patterns. Those patterns are the regularity among odd numbers for eliminating the composite number such as Sieve of Eratosthenes. Those patterns shows the regularity among the odd numbers except 2. Those patterns may be used to determine which odd numbers are prime numbers. This pattern may reduce the computation time to find prime numbers in some number range.
Category: Number Theory
[3588] viXra:2512.0030 [pdf] submitted on 2025-12-07 20:22:13
Authors: Ansh Mathur
Comments: 8 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper introduces the Universal Divisibility Framework (UDF), a comprehensive mathematical theory that extends the classical notion of divisibility from integers to rationals, reals, and complex numbers. The framework is built upon the Universal Divisibility Function $d(a, b, c) = lfloor a/c floor (b bmod c) - lfloor b/c floor (a bmod c)$, which provides a unified criterion for divisibility across multiple number systems. We establish the Universal Divisibility Theorem, proving that for $a, b in mathbb{R}$ with $b eq 0$, and an integer $c$ satisfying $lfloor b/c floor = pm 1$, we have $b mid a$ if and only if $d(a, b, c) equiv 0 pmod{b}$. This framework not only recovers all classical integer divisibility rules as special cases but also eliminates false positives that arise when traditional rules are naively extended to non-integer domains. We provide explicit divisibility formulas for numbers 1—1000, demonstrate applications to Diophantine equations and matrix algebras, and discuss implications for computational number theory and cryptography.
Category: Number Theory
[3587] viXra:2511.0148 [pdf] submitted on 2025-11-30 02:24:38
Authors: Durga Shankar Akodia
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We study integer solutions to the exponential Diophantine equation x^2 + k = 2^n where k and n are primes. We prove that for any solution with n >= 2, x must be odd, and k must be an odd prime. Furthermore, we establish the strict congruence k = 7 (mod 8) for all solutions with n >= 3. We identify a trivial family of solutions corresponding to Mersenne primes (x=1) and demonstrate the existence of non-trivial solutions for x > 1. Computational evidence is presented for 9 out of 11 prime values of n <= 31, revealing 16 distinct non-trivial solutions. We propose the Akodia Conjecture concerning the infinitude of such non-trivial solutions
Category: Number Theory
[3586] viXra:2511.0136 [pdf] submitted on 2025-11-26 11:16:02
Authors: Dmitri Martila
Comments: 2 Pages.
I am writing a shortest proof of the Riemann Hypothesis using Wang's paper as a starting point.
Category: Number Theory
[3585] viXra:2511.0133 [pdf] submitted on 2025-11-26 23:29:53
Authors: Ryan Hackbarth
Comments: 5 Pages.
In this paper, I utilize Euler's derivation of the Product For the Zeta function to produce a similar Product Formula for the Dirichleta Eta Function. I then examine how this formula relates to the critical line and the zeros of the Zeta Function
Category: Number Theory
[3584] viXra:2511.0130 [pdf] submitted on 2025-11-26 01:26:58
Authors: Ahcene Ait Saadi
Comments: 7 Pages. (Note by viXra Admin: Further repetition may not be accepted; for the last time, please cite and list scientific references!)
This document deals with enigmas related to the numbers 17 and 19, by presenting equations involving squares of natural integers. It is composed of two parts. 1-Enigmas of the numbers 17 and 19[;] 2-Enigma of the number 17
Category: Number Theory
[3583] viXra:2511.0124 [pdf] submitted on 2025-11-24 22:07:52
Authors: Cesar A. P. Correa
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a heuristic approach to the validity of the Riemann Hypothesis, utilizing techniques from Harmonic Analysis and Partial Differential Equations. By decomposing the Dirichlet series using integer-part functions, an oscillatory discrepancy term generated by the discrete nature of the summands is isolated. Modeling this term via Fourier series and subjecting it to the Laplacian operator in the complex plane, it is demonstrated that the condition for the annihilation of the function $zeta(s)$ requires an equilibrium of magnitudes in the second-order partial derivatives. Analytical results indicate that such equilibrium is unstable for $text{Re}(s) eq 1/2$, providing strong theoretical evidence in favor of Riemann's original conjecture.
Category: Number Theory
[3582] viXra:2511.0118 [pdf] submitted on 2025-11-24 01:49:22
Authors: Quency Nixon
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
This paper presents a deterministic, structural proof of the Twin Prime Conjecture, moving beyond traditional probabilistic models. We introduce the "Midpoint Generator" (Mn = Pn/2), a geometric center of gravity that necessitates the existence of Twin Prime pairs at every scale of the number line. We define the "Recursive Binary Descent," an algorithm that mechanically links candidate pairs at infinite scales to verified Twin Primes in a finite "Safe Haven." Finally, using a Proof by Contradiction, we demonstrate that the assumption of a finite number of Twin Prime pairs creates a structural paradox, thereby proving that the set of TwinPrime pairs must be infinite.
Category: Number Theory
[3581] viXra:2511.0114 [pdf] submitted on 2025-11-23 00:25:22
Authors: Michael Spencer
Comments: 63 Pages. (Note by viXra Admin: Please cite and list scientific references of other authorities besides self-citations and submit article written with AI assistance to ai.viXra.org)
This paper presents a complete arithmetic resolution of the Collatz Conjecture by separating its structure into two complementary components: a finite residue and phase transition system, and a global affine counting framework. The reverse Collatz step is shown to act only on the two live odd residue classes, and every valid reverse exponent produces a predictable affine expansion whose inverse matches the expected dyadic frequency. From this, every odd integer belongs to exactly one uniquely defined dyadic slice, forming a disjoint partition of all odd numbers. Independently, a zero-state index reveals that every live odd number also generates a unique sequence obtained by repeatedly applying the transformation four times the number plus one, and these sequences likewise partition the odd integers without overlap. The two partitions are proven to be identical, establishing a single global organizational structure for the entire Collatz map. Because the forward and reverse processes are locked to each other and the residue-phase system is finite, every forward trajectory follows one non-branching path that must ultimately return to one. This eliminates the possibility of infinite growth or nontrivial cycles. All structural components, classifications, and counting methods are original to this work and together provide a fully closed arithmetic description of the Collatz dynamics.
Category: Number Theory
[3580] viXra:2511.0111 [pdf] submitted on 2025-11-22 01:10:10
Authors: F. F. Martinez Gamo
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present computational evidence for a novel spectral characterization of prime numbers through the Laplacian eigenvalues of Paley-type graphs. For integers n ≡ 1 (mod 4), we demonstrate that the second smallest Laplacian eigenvalue λu2082 of the graph constructed from quadratic residues modulo n satisfies λu2082(G) = (n - √n)/2 if and only if n is prime, with numerical precision limited only by floating-point accuracy (~10u207b¹u2075). Composite numbers exhibit substantial deviation from this formula, with gaps ranging from 3 to over 60 for n < 300. Statistical analysis over 29 primes and 30 composites shows a separation ratio exceeding 10¹u2075 between prime and composite gap magnitudes. This result establishes a connection between number-theoretic primality and graph spectral properties, with implications for understanding the algebraic structure of finite fields versus rings with zero divisors.
Category: Number Theory
[3579] viXra:2511.0103 [pdf] submitted on 2025-11-21 00:04:18
Authors: Shinsuke Hamaji
Comments: 12 Pages. [Submitted to a journal]; Zenodo: https://doi.org/10.5281/zen (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Abstract: This paper identifies the fundamental difficulty of the long-unresolved Collatz Conjecture as stemming from an unconscious self-limitation in conventional mathematics—specifically, a structural ``lack of collaboration'' between the linear definition of natural numbers (Peano Axioms) and non-linear structural analysis (Cantor's Set Theory). We propose that, to solve a non-linear Halting Problem like the Collatz Conjecture, the proof must prioritize the structural fundamental of ``symmetry of the start and stop point ($1$)'' instead of adhering to linear methods. Specifically, we resolve the ``bijective definition deficit'' of the mapping existing between specific sub-patterns (e.g., $4n+1 leftrightarrow 3n+1$) derived from the Collatz operation, by using Cantor's dimensional expansion pairing function. This method reconstructs the Collatz operation as a closed, bijective structure centered at $1$, structurally and completely excluding the possibility of cycles other than $1$ and divergence to infinity. This represents a structural solution that, by integrating the Peano Axioms and Cantor's Set Theory, rigorously guarantees the global stability of the Collatz infinite tree for the first time.Keywords: Collatz, Tree Equivalence Theorem, Peano's Successor Function, Cantor's Pairing Function.MSC 2020: 03D50, 11B83.
Category: Number Theory
[3578] viXra:2511.0096 [pdf] submitted on 2025-11-20 00:16:58
Authors: Christoper Muoki Mututu
Comments: 10 Pages. (Note by viXra Admin: Please cite and list scientific references)
Prime numbers have fascinated mathematicians for centuries due to their inherent unpredictability and fundamental role in number theory. Despite extensive research into their distribution and patterns, primes continue to surprise and challenge scholars. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, is a prime example of this unpredictability. This paper introduces a newly discovered class of prime numbers which under two distinct alternating gap sequences — 2,4,2 and 12,14,12 - predict the occurrence of six additional primes, either forward or in reverse, a phenomenon previously unknown in the study of prime numbers. This work offers not only a novel approach to prime generation but also introduces the idea that primes themselves can act as building blocks for other primes, leading to new methods of understanding prime distribution and its profound implications for both theoretical and applied mathematics.
Category: Number Theory
[3577] viXra:2511.0093 [pdf] submitted on 2025-11-18 22:51:56
Authors: Immense Raj Subedi
Comments: 12 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a novel approach to the Collatz conjecture by focusing on the subset of natural numbers expressed in the form 12n − 4. By analyzing the algebraic mappings and trajectories of these numbers under the Collatz function, we demonstrate that their sequences remain within this form and exhibit a strictly decreasing behavior. We establish that the transformations lead to a pipeline of values that map back to smaller terms of the same form, thereby ensuring infinite descent and convergence to 1. Since every natural number eventually reaches an odd number, and the odd numbers correspond to this subset via our mapping, the results imply convergence of all natural numbers to 1. The methodology combines explicit algebraic mappings with inequalities to show the completeness of coverage and the absence of nontrivial cycles, providing strong evidence towards a proof of the conjecture
Category: Number Theory
[3576] viXra:2511.0068 [pdf] submitted on 2025-11-14 22:21:36
Authors: Muhammad Razzaq Aman Wattoo
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper proposes a unified theoretical framework integrating the Uu2011Water Continuum Theory with the Golden Ratio (ϕ) and the Fibonacci Sequence. While Uu2011Water posits the universe as a continuous medium rather than discrete particles or relativistic distortions, the Golden Ratio represents external proportional expansion, and Fibonacci numbers represent internal developmental sequences governing growth. We formalize a mathematical fusion showing how Fibonacci structures propagate through the Uu2011Water continuum to generate ϕ-based expansions: "Flow"_n=F_n⋅ϕ^n,F_(n+2)=F_(n+1)+F_n,ϕ=(1+√5)/2 [1-3] Advanced modeling includes polar Fibonacci spirals, continuum-based differential equations, and generating functions, establishing a triadic system: internal progression (Fibonacci), external proportionality (ϕ), and medium-based continuity (Uu2011Water).
Category: Number Theory
[3575] viXra:2511.0050 [pdf] submitted on 2025-11-11 21:17:49
Authors: Manikandan Karunanidhi
Comments: 4 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This work introduces a new family of natural numbers for which the sum of two successive powers, beginning with an even exponent, results in a perfect square. The identity is derived from the condition where the base plus one equals a perfect square. A formal proof is presented, along with a recurrence relation and supporting computational evidence in tabular form. This study contributes to the field of number theory by highlighting an unexpected relationship between exponential expressions and perfect squares.
Category: Number Theory
[3574] viXra:2511.0040 [pdf] submitted on 2025-11-10 20:59:22
Authors: J. Kuzmanis
Comments: 23 Pages.
Primary abc-triples, formed by the set of roots for the generalized Pell’s equations x^2 -Dy^2 = +/-N (with N [larger than] 2), induce formation of secondary abc-triples in the set of roots for equations x^2 - Dy^2 = N^2.
Category: Number Theory
[3573] viXra:2511.0035 [pdf] submitted on 2025-11-08 16:14:33
Authors: Felipe Wescoup
Comments: 9 Pages.
This paper presents a method for generating lists of prime numbers. The algorithm presentedcalculates the set of all composite (non-prime) numbers up to a given limit and the set of primes is subsequently defined as its complement. The method is demonstrated through a JavaScript implementation, which is evolved over four iterative levels increasing optimization. Level 1 presents the core mathematical concept. Levels 2 and 3 include logical assumptions thatimprove efficiency. Level 4 is capable of calculating all primes up to 20 million in seconds within a standard web browser. This paper is supplemented by a public GitHub repository containing the complete operational code.
Category: Number Theory
[3572] viXra:2511.0025 [pdf] submitted on 2025-11-07 01:30:00
Authors: Colm Gallagher
Comments: 11 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
A deterministic arithmetic reformulation of the mod 6 lattice revealing geometric symmetry in the distribution of primes. Rotational symmetry when applied to these residuals visits all of the non-prime numbers stepwise, which we refer to as "hops", that generates a deterministic arithmetic framework that reproduces the sequence of primes and their gaps. We present a framework for understanding the distribution of primes using modular arithmetic and iterative hop sequences. Visual patterns such as the Ulam spiral are shownto arise naturally from rotational symmetries within this framework. We provide both anintuitive explanation and a formal arithmetic treatment that reproduces the sequence ofprimes and their gaps. As first noted by Ulam and popularized by Gardner, the arrangement of integersin a spiral lattice reveals that prime numbers tend to cluster along diagonal lines. However,this observation alone does not explain the *mechanism* of the clustering. The hop-based interpretation offers a natural explanation: primes occupy loci defined by arithmetic propagation rather than arbitrary geometric coincidence.
Category: Number Theory
[3571] viXra:2511.0024 [pdf] submitted on 2025-11-07 01:27:01
Authors: Satya Das
Comments: 23 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We establish a structural correspondence between the Collatz map and the signed Jacobsthal numbers, providing an arithmetic reformulation of the 3x + 1 problem. By representing Collatz iterations through powers of signed Jacobsthal numbers, we derivenecessary and sucient conditions for the existence of cycles and for the validity of the coecient stopping time conjecture. This formulation translates the combinatorial dynamics of the Collatz map into explicit number-theoretic identities, revealing an underlying algebraic framework that connects iteration, recurrence, and integrality. The results suggest a pathway toward analyzing the conjecture through the intrinsicarithmetic structure of the signed Jacobsthal numbers.
Category: Number Theory
[3570] viXra:2511.0016 [pdf] submitted on 2025-11-06 02:27:33
Authors: WenBin Hu
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper proposes a set of logical operation rules for sequences and formulates a generation rule for difference-free sequences that satisfy the operations. The "difference-free sequence" in this paper refers to a sequence where the difference between any two arbitrary numbers within the sequence is not equal to any number in other sequences.Common generation rules for difference-free sequences include:The new term of the sequence satisfies a_(n+1)>2a_nComputer-generated sequences based on the greedy algorithm
Category: Number Theory
[3569] viXra:2510.0154 [pdf] submitted on 2025-10-31 16:30:15
Authors: Fian Qnoz
Comments: 12 Pages. 37 figures.
The exploration of incomplete magic square of squares, whose being fully working magic square with less than 9 square entries, leads to the extensive use of Brahmagupta-Fibonacci identity. By only taking account of primitive (irreducible) entries, the umbrella term Brahmagupta’s Abacial Slice of Irreducible Calamari (BASIC) magical marine square was adopted. Interesting varieties ranging from congruent elliptic curves up to the affine variety A6 along with the K3 surface of degree 8 were encountered when one considers the birational model of Magicmare.
Category: Number Theory
[3568] viXra:2510.0135 [pdf] submitted on 2025-10-27 23:54:29
Authors: Zihang Chen
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper mainly studies the new analytic continuation method of the Riemann zeta function and the properties of the zeta function after analytic continuation, and explores the laws of its zeros based on these properties. The author conducts the analytic continuation of the Riemann zeta function via the Euler-Maclaurin formula, and finds that it has a close connection with the Bernoulli recurrence formula through analysis. Through special construction and mutual transformation with the gamma function, a special symmetric relationship of the zeta function is discovered, thereby identifying the distribution law of the non-trivial zeros of the zeta function. The new methods and new ideas used in this paper include the new analytic continuation form of the zeta function, the special construction method, and so on.
Category: Number Theory
[3567] viXra:2510.0129 [pdf] submitted on 2025-10-26 23:04:55
Authors: Pravin Kumar Mishra
Comments: 10 Pages. (Note by viXra Admin: Please cite listed scientific reference)
This paper presents a series of theorems and corollaries in two sections. The 2nd section outlines a method for verifying the existence of prime numbers within specific intervals. Postulates 1 and 2 establish a methodology to verify the existence of primes in the specific intervals shown in Theorem 1 and its corollary and Theorem 2. The 3rd section outlines a prime indicator function that generates all primes sequentially. The construction of this indicator begins from Theorem 3, and Theorem 4 provides insights on the sum of all odd composite numbers, and its corollary produces a prime indicator supported by an Illustration of a few first numbers. This work provides new insights into how to use elementary principles and methods.
Category: Number Theory
[3566] viXra:2510.0127 [pdf] submitted on 2025-10-26 22:51:54
Authors: Jean-Yves Boulay
Comments: 47 Pages.
Grounded in a novel mathematical framework, this study partitions the set of whole numbers (ℕ0) into four distinct hierarchical classes. A key innovation is the definition of Ultimate Numbers—the union of the prime numbers with zero and one—which resolves classic conceptual limitations. Three further subsets, representing increasing degrees of numerical complexity, are subsequently defined by the initial distinction between ultimate and non-ultimate numbers within ℕ0. The structural interaction among these four classes yields unique arithmetic arrangements in their initial distribution, most notably revealing an exact and recurring 3:2 ratio.
Category: Number Theory
[3565] viXra:2510.0073 [pdf] submitted on 2025-10-14 20:45:29
Authors: Khalid ibraheem Al-Ibraheem
Comments: 14 Pages. There is an error in scheme 1, I will fix it in the second version.
This paper presents a complete and novel proof of the Collatz conjecture.The proof is built upon a fundamental reduction showing that convergence for all positiveintegers follows from convergence for all odd integers. We then introduce a novel ternarypartition of the set of all odd integers into three mutually exclusive and exhaustivesets B, C, and D.A pivotal element of the proof is the introduction and detailed examination of the setV = {5 + 12 .n | n ≥ 0} .
Category: Number Theory
[3564] viXra:2510.0061 [pdf] submitted on 2025-10-13 20:19:05
Authors: Chandhru Srinivasan
Comments: 11 Pages. (Note by viXra Admin: Author name is required in the article; please submit article written with AI assistance to ai.viXra.org)
I report an empirical derived and theoretically motivated analysis of modular patterns in composite integers, with a focus on semiprimes. For any odd semiprime N(possibly all N ∈ ℤ), the results indicate the existence of N-1congruences of the form p+q≡r(modm), where p and q are factors of N, m∈{2,u2026,N-1} as 1 and N are trivial and always N ≡0 mod(1 or N) , and each residue r belongs to a restricted, well-structured subset R_m. Empirical experiments suggest that these residue constraints are non-random, deterministic and encodes all the necessary information about the factor pair (p,q). I formalize this observation as a conjecture and provide preliminary reasoning for its generality. These results point toward a potentially deterministic structure in the modular representation of factor sums, and potentially speedup the factorisation of any N and offering a new perspective on the arithmetic properties of semiprimes and composite numbers. I invite further mathematical verification and formalization.
Category: Number Theory
[3563] viXra:2510.0052 [pdf] submitted on 2025-10-10 20:20:23
Authors: Hashem Sazegar
Comments: 5 Pages.
Oppermann’s conjecture states that for every positive integer n, there exists at least one prime number between n 2 and n 2 + n. Priorto this, Legendre had conjectured that there is always at least one prime number between n2 and (n + 1)2 . In this paper, we not onlyclaim to prove Oppermann’s conjecture but also propose a smaller interval, asserting that there exists at least one prime between n 2 andn 2 + n/2.
Category: Number Theory
[3562] viXra:2510.0051 [pdf] submitted on 2025-10-09 20:57:18
Authors: Theophilus Agama
Comments: 10 Pages. (Note by viXra Admin: Frequent/incessant submissions of highly speculative/abstract articles will not be accepted)
We prove an extension of the lower bound due to Schonhage on addition chains.
Category: Number Theory
[3561] viXra:2510.0022 [pdf] submitted on 2025-10-05 17:27:40
Authors: Islem Ghaffor
Comments: 1 Page.
In this paper we prove Collatz conjecture by giving an equivalent formulation of the shortcut Collatz sequence.
Category: Number Theory
[3560] viXra:2509.0158 [pdf] submitted on 2025-09-30 00:00:47
Authors: Theophilus Agama
Comments: 3 Pages.
We develop a criterion for an addition chain to have low energy and pose a related classification problem.
Category: Number Theory
[3559] viXra:2509.0155 [pdf] submitted on 2025-09-30 20:57:24
Authors: Fian Qnoz
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references)
Diophantine equation of the form a^3+b^3 = 2(2^5 - c^3)relates to the quadratic equation 1 + 4n + 4n^2 which further relates to 1x^3 + 4y^3 + 4z^3 = 512 whose parametric solution for x is exactly is the twice of the square of odd number > 1, i.e. 2(2n+1)^2. Considering phi (sum of the odd divisors of 2(2n+1)^2) is exactly equal to phi (sum of the even divisors of 2(2n+1)^2), some interesting properties involving Piltz functions and Jordan totients were conjectured.
Category: Number Theory
[3558] viXra:2509.0152 [pdf] submitted on 2025-09-30 23:19:34
Authors: Theophilus Agama
Comments: 5 Pages.
Let E(n): 1 = s_0 smaller than s_1 = 2 smaller than u2026 smaller than s_h = n, with h on the order of n to the power 1 minus epsilon (for some small positive epsilon), be an addition chain leading to n. We develop a heuristic for the least number of primes in an addition chain for all sufficiently large targets n.
Category: Number Theory
[3557] viXra:2509.0146 [pdf] submitted on 2025-09-29 02:09:09
Authors: Aditya Bagchi
Comments: 50 Pages.
This paper presents a complete proof of the conjecture by demonstrating that both of the counterargument scenarios are mathematically impossible. Our approach is deterministic and algebraic, built upon two analytical frameworks. The first is a Perturbation Model used to prove the non-existence of non-trivial integer cycles. The second is a Modular Loop Framework used to prove the non-existence of divergent trajectories. The theoretical claims of both frameworks are supported by extensive, reproducible computational evidence.
Category: Number Theory
[3556] viXra:2509.0133 [pdf] submitted on 2025-09-25 09:02:17
Authors: Jérôme Chauvet
Comments: 4 Pages.
We adress here the TPC (Twin Primes Conjecture). Considering the unique growing list of twin primes as elements of a dynamical system with invariant transformation law on, we infer a lower boundary for their density along the real axis incompatible with their finiteness. Although non constructive, since we do not prove a general formula for finding twin primes pairs arbitrarily great, this proof relies for closure on the tertium non datur principle with regard to their minimal density, and thus their cardinality at infinity
Category: Number Theory
[3555] viXra:2509.0123 [pdf] submitted on 2025-09-21 03:05:28
Authors: Theophilus Agama
Comments: 14 Pages.
Let E(n): 1 = su2080 < su2081 = 2 < u2026 < s_h = n be an addition chain leading to n in [2^m, 2^{m+1}). We study the distribution of the logarithmic partial sum ∑ log(s_i) on maximal consecutive steps of a given type.
Category: Number Theory
[3554] viXra:2509.0114 [pdf] submitted on 2025-09-19 09:25:21
Authors: Marko V. Jankovic
Comments: 4 Pages.
t has been recently explained that Riemann rearrangement theorem is wrong [1], and that it has never been correctly proved. In [1] was demonstrated, on a famous example, that it is not the rearrangement of the elements, but rather the omission of elements of the conditionally convergent series, that would lead to a different summation result.The example that was used in [1] does not strictly follow the Riemann rearrangement method that is proposed in his theorem. It was correctly detected by Google's AI module. In this paper an example that follows the Riemann rearrangement method is going to be presented and again, it is going to be explained that the reason the "rearranged"series has a different sum is the omission of the infinite number of elements of the original series. Generally speaking, the rearrangement method has no critical impact on the summation result — the summation result depends on the sum of elements that are not included in the sum, and that is very simple to understand
Category: Number Theory
[3553] viXra:2509.0102 [pdf] submitted on 2025-09-17 14:24:04
Authors: Dmitri Martila
Comments: 3 Pages.
The theta(x) < x + 0.5 ln x implies new major results, i.e., proofs of many conjectures.
Category: Number Theory
[3552] viXra:2509.0097 [pdf] submitted on 2025-09-16 17:05:36
Authors: Kuan Peng
Comments: 10 Pages. (impropriate materials removed by viXra Admin)
In this article we have created the table that classifies all primitive triples, shown some properties of basic triples and discussed about the use of primitive Pythagorean triples in cryptography.
Category: Number Theory
[3551] viXra:2509.0094 [pdf] submitted on 2025-09-15 08:30:51
Authors: Theophilus Agama
Comments: 9 Pages.
We prove that a certain class of infinite sequences whose finite truncation is an addition chain must have arbitrarily large gaps between their consecutive terms. This result is generic and can be applied to particular known infinite sequences with this property.
Category: Number Theory
[3550] viXra:2509.0091 [pdf] submitted on 2025-09-15 20:02:10
Authors: Uggala Guru Jaganath
Comments: 7 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper introduces Jagan’s Primes, a novel recursive modular equation for the deterministic generation of prime numbers in exact order. Unlike traditional sieve-based or probabilistic methods, this framework constructs primes through a self-referential arithmetic logic that isolates non-prime residues via modular shielding. The equation operates without primality testing, leveraging recursive congruence relations to build the prime sequence from first principles. The approach offers a new perspective on prime enumeration, demonstrating that primes can emerge as a consequence of structural recursion rather than exclusion. This work contributes to the foundations of computational number theory and opens pathways for algorithmic applications in cryptography, mathematical linguistics, and symbolic modeling.
Category: Number Theory
[3549] viXra:2509.0087 [pdf] submitted on 2025-09-15 20:31:42
Authors: Yonatan Zilpa
Comments: 17 Pages. (Note by viXra Admin: Please cite all listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper explores the zeros of symmetric analytic functions, focusing on the Riemann zeta function. Using the Abel-Plana formula and an auxiliary function, we investigate their distribution. Our approach reformulates the problem algebraically and employs a proof by contradiction. We demonstrate this approach by applying it to the Riemann zeta function.
Category: Number Theory
[3548] viXra:2509.0073 [pdf] submitted on 2025-09-12 16:35:30
Authors: Sugahara Jotaro
Comments: 8 Pages. (Note by viXra Admin: An abstract in the article is required; please cite listed scientific referenes and submit article written with AI assistance to ai.viXra.org)
[This paper explores the a rigorous equivalence between Phase Drift and the Riemann Hypothesis]
Category: Number Theory
[3547] viXra:2509.0055 [pdf] submitted on 2025-09-09 12:19:39
Authors: Mar Detic
Comments: 7 Pages.
This paper provides a comprehensive analysis of the Diophantine equations 2m2+2m =yn andm(m+2) =yn forintegersm,y ≥ 0andn ≥ 2. Wedemonstrate that the first equation has infinitely many solutions for n = 2 (via a Pell equation) and only the trivial solution for n ≥ 3 (by Erd˝os—Selfridge), while the second has no nontrivial solutions for any n ≥ 2. We explore connections to Fermat’s Last Theorem, the Beal Conjecture, and the ABC Conjecture. Additionally, we show that for odd m = 2k+1, the equation m(m+2) = yn becomes 4(k+1)2−1 = yn, connecting it to arithmetic progressions and Pell-type equations. We demonstrate that attempts to express these equations in Beal form fail, and we highlight the role of discriminants and factorization in determining the existence of solutions
Category: Number Theory
[3546] viXra:2509.0037 [pdf] submitted on 2025-09-06 22:46:10
Authors: Rusin Danilo Olegovich
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
We present a complete structural proof of the Riemann Hypothesis, based on the interplay between the canonical well-ordering of nontrivial zeros and the symmetry imposed by the functional equation. Working from three established analytic properties of the Riemann zeta function — discreteness of zeros, confinement to the critical strip, and functional equation symmetry — we construct a proof that reduces the Riemann Hypothesis to a purely combinatorial statement about order and symmetry. We prove the Critical Gap Theorem: if a zero off the critical line exists, the first such zero (under the canonical ordering) must lie to the left of the critical line, forcing its functional equation partner to appear later in the ordering. This leads to a contradiction unless no such zero exists. The result is a logically complete, structurally elegant proof of the Riemann Hypothesis requiring no advanced analytic estimates — only classical properties known since the 19th century.
Category: Number Theory
[3545] viXra:2509.0033 [pdf] submitted on 2025-09-05 16:30:58
Authors: Rayan Bhuttoo
Comments: 7 Pages. License: CC BY-NC-ND 4.0 (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We demonstrate that prime numbers are precisely the indecomposable elements under a novel group operation defined on a circle, providing a geometric characterization of primality.By projecting integers onto a circle via the mapping θn = arccos(n/R), we show thatprimes exhibit intense clustering at the endpoints of the diameter, while composite numbers distribute uniformly. We formalize this observation by defining an angular density function F (n) that vanishes if and only if n is prime, with a rigorous proof based on the PrimeNumber Theorem. Furthermore, we analyze the Fourier spectrum of the prime distribution,revealing a distinct high-frequency signature. Finally, we conjecture connections betweenthis harmonic signature and the nontrivial zeros of the Riemann zeta function, suggesting anew approach to understanding prime distribution through geometric and harmonic analysis.
Category: Number Theory
[3544] viXra:2509.0008 [pdf] submitted on 2025-09-01 22:53:31
Authors: Ahcene Ait Saadi
Comments: 11 Pages. (Note by viXra Admin: Further repetition may not be accepted; please cite and list scientific references in the article)
This document is entitled (mysteries of prime numbers and magic matrices) Explores the relationships between prime numbers and special matrices. The main objective is to use these matrices to form triples of prime numbers and to establish mathematical conjectures. This work may have common mathematical relationships with the Golbach conjecture and Collatz conjecture. The research is only at is beginnings, I hope that young researchers will be interested in it, and why not draw mathematical theory’s from it.Key words: Prime numbers; Matrices; System of equality, Some of square of prime numbers.
Category: Number Theory
[3543] viXra:2509.0005 [pdf] submitted on 2025-09-01 19:25:13
Authors: Jabari Zakiya
Comments: 14 Pages.
Various bounds on p, such as Bertrand’s Postulate and Legendre’s Conjecture, propose regions around n that have at at least one prime within them. Using Prime Generator Theory, I show more precise symmetric bounds on p, such that for n a primeexists symmetrically within a distance of n^(1/2) below and above it. That is to say, a prime exists for: n — n^(1/2) < p < n and n < p < n + n^(1/2).
Category: Number Theory
[3542] viXra:2508.0189 [pdf] submitted on 2025-08-31 03:18:24
Authors: Kurmet Sultan
Comments: 2 Pages.
The article presents theorems and their proofs, which in turn prove the infinity of prime twins, i.e. provides a solution to the second Landau problem.
Category: Number Theory
[3541] viXra:2508.0187 [pdf] submitted on 2025-08-31 20:19:11
Authors: Rayan Bhuttoo
Comments: 6 Pages. License: CC BY-NC-ND 4.0 (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
The Collatz conjecture remains a formidable open problem in number theory. This paperpresents a novel reformulation of the Collatz function, T(n), demonstrating that it is equivalent to the operation (n · (n + 1)) mod (n + (n + 1)). This identity transforms the traditionally piecewise-defined map into a single, unified algebraic operation performed within the quotient ring Z/(2n + 1)Z. This perspective intrinsically connects the conjecture to the properties of consecutive integers and the structure of modular rings. Furthermore, it provides a natural geometric interpretation of the iteration process. This reformulation does not constitute a proofof the conjecture but offers a new and powerful framework that opens new avenues for attackingthe problem through ring theory, analysis, and geometry.
Category: Number Theory
[3540] viXra:2508.0185 [pdf] submitted on 2025-08-31 20:15:52
Authors: Wilson Gomes
Comments: 8 Pages. License: CC BY-NC-ND 4.0 (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
We present the first complete and rigorous proof of Polignac’s Conjecture using a novel unified spectral approach that combines Tsallis nonextensive statistics, Hilbert space theory, and advanced sieve methods. By reformulating the prime gap sequence in a weighted Hilbert spacewith memory effects, we derive a fundamental spectral identity connecting gap persistence to zeta functions. Through rigorous analysis of pivot operators with proven exponential mixing properties and explicit computation of sieve-theoretic bounds for each even gap, we establish the infinitude of every fixed even gap size. The proof is validated by extensive numerical computations up to 10^15 and provides explicit constants for gaps n = 2, 4, 6, 8, 10.
Category: Number Theory
[3539] viXra:2508.0170 [pdf] submitted on 2025-08-28 15:40:18
Authors: Abdelhay Benmoussa
Comments: 13 Pages. Submitted to a journal
We study the integral analog of the operator (left(x frac{d}{dx}right)^n), obtained by replacing differentiation with integration. We prove that the resulting operator admits an expansion in powers of the integration operator with coefficients given by the Bessel numbers of the second kind ({B(n,k)}) (OEIS seqnum{A122848}), leading to new explicit formulas and revealing a fundamental role of Bessel numbers in the structure of certain integral operators.
Category: Number Theory
[3538] viXra:2508.0168 [pdf] submitted on 2025-08-28 20:31:46
Authors: Guiffra Patrick
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We explore the application of the **Discrete Fourier Transform (DFT)** to a class of modular wave functions. For an integer q ≥ 2, a Dirichlet character χ modulo q, and an integer p coprime to q, we introduce the modular wave function ψp(x) = χ(p)exp[(i2πp_1/q)x, where p−1 is the modular inverse of p modulo q.We rigorously demonstrate that the DFT of ψp(x) is a **Kronecker delta peak** with a value of χ(p) √q, located precisely at the frequency k = p^−1 (mod q), and zero everywhere else. This result illustrates a direct and elegant connection between modular inverses and spectral analysis, showing how arithmetic structures can be encoded and detected using signal processing tools.
Category: Number Theory
[3537] viXra:2508.0166 [pdf] submitted on 2025-08-28 20:23:54
Authors: Carlos Alejandro Chiappini
Comments: 3 Pages. carloschiappini@gmail.com (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We learned in school that 2n corresponds to even numbers and that 2n-1 corresponds to odd numbers. In this document, we'll see a formula for periodic numbers, almost as simple as the first two.
Category: Number Theory
[3536] viXra:2508.0165 [pdf] submitted on 2025-08-26 23:41:30
Authors: Teo Banica
Comments: 400 Pages.
This is an introduction to numbers, fractions, percentages and arithmetic. We first discuss what can be done with integers and their quotients, namely basic arithmetic, a look into prime numbers, all sorts of counting results, and with a look into percentages and basic probability too. We then upgrade our knowledge by introducing the real numbers, and exploring what can be done with them, in relation with number theory questions. Then we further upgrade our methods, by introducing and using the complex numbers. Finally, we provide an introduction to modern number theory.
Category: Number Theory
[3535] viXra:2508.0161 [pdf] submitted on 2025-08-27 20:14:28
Authors: Jim Rock
Comments: 23 Pages. (Note by viXra Admin: Further repetition will not be accepted and please submit article written with AI assistance to ai.viXra.org)
Collatz sequences originate from dividing an even number by two until an odd number is obtained, followed by multiplication by three and an increment of one to yield an even number. The Collatz conjecture posits that the repeated application of this process inevitably results in the number one. The Collatz conjecture holds true for every number tested, but no general method has been found to prove that it is true for all positive integers. We introduce a new methodology: the binary series. In conjunction with mathematical induction, this new methodology provides a more general method of testing positive integers for properties that cannot be established by induction alone. We partition the positive integers into distinct subsets. The binary series allows us to use geometric series that sum to one (100%) to show that all natural numbers satisfy the Collatz conjecture. This new methodology eliminates the need to test every integer and provides a general method of proof for the Collatz conjecture.
Category: Number Theory
[3534] viXra:2508.0151 [pdf] submitted on 2025-08-25 06:15:17
Authors: Kurmet Sultan
Comments: 2 Pages.
A proof of Legendre’s conjecture is obtained by establishing the following regularity: in any interval between the squares of two consecutive positive integers, the number of odd integers of the form 6a∓16a mp 1 strictly exceeds the number of integers of the same form that can be represented as6b∓1=(6m∓1)(6(m+x)∓1).6b mp 1 = (6m mp 1)(6(m+x) mp 1).
Category: Number Theory
[3533] viXra:2508.0143 [pdf] submitted on 2025-08-23 22:38:38
Authors: Jordan Gidman
Comments: 13 Pages. https://github.com/CoreTheoretics/Riemann-Scripts (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present computational evidence for a novel framework suggesting that the critical line σ = 1 2 in the Riemann zeta function exhibits mathematical attractor properties for zero formation and gap prediction. Through systematic analysis of Dirichlet partial sum approximations across verified nontrivial zeros, we demonstrate that metrics measuring zero-formation dynamics consistently peak at σ = 1 2 with remarkable stability across parameter variations. Our bootstrap resampling analysis yields a zero-collapse attractor location of σ = 0.500000±0.000000 across 20 independent trials (n=500 each). We introduce a gap-confidence prediction framework achieving 50% success rates within calibrated uncertainty bounds while maintaining 1.23% relative prediction accuracy. Comparative analysis between ζ(s) and 1/ζ(s) reveals identical predictive behavior, suggesting functional invariance of gap patterns. These findings support a conjecture that the Riemann Hypothesis emerges from fundamental attractor dynamics rather than coincidental zero placement, providing new computational approaches to critical line analysis.
Category: Number Theory
[3532] viXra:2508.0128 [pdf] submitted on 2025-08-19 02:10:12
Authors: Gustavo García, Oscar Melchor
Comments: 3 Pages.
Assuming the validity of Goldbach’s strong conjecture, we derive a family of inductive consequences concerning the representation of natural numbers as sums of primes. As context, we recall the classical theorems of Vinogradov and Chen, which constitute the most celebrated unconditional advances toward Goldbach’s conjecture. We then present the inductive statements that would follow immediately if Goldbach’s conjecture were true. A proposed elementary proof of Goldbach’s conjecture has been provided by the author at https://vixra.org/author/gustavo_garcia
Category: Number Theory
[3531] viXra:2508.0115 [pdf] submitted on 2025-08-18 20:49:58
Authors: Ammar Hamdous
Comments: 29 Pages. CC-BY-NC 4.0 (Note by viXra Admin: Please cite and list scientific references)
The Collatz conjecture, first proposed by Lothar Collatz in 1937, has captivated generations of mathematicians due to its deceptive simplicity and its enduring resistance to proof. Also referred to as the 3n+1 problem, the Syracuse problem, the Ulam conjecture, or the Hailstone sequence, it has spread informally across academic communities, often through oral tradition and recreational mathematics. Its basic rule can be explained to a child, yetits resolution has defied the most brilliant minds in mathematics. As Shizuo Kakutani noted in 1960, "For about a month everyone at Yale worked on it, with no result... A joke was made that this problem was part of a conspiracy toslow down mathematical research in the U.S." Paul Erdős, in 1983, famously declared that "Mathematics is not yet ready for such questions." More recently, in 2010, Jeffrey Lagarias described it as "an extraordinarily difficult problem, completely out of reach of present day mathematics. The conjecture sits atthe intersection of several mathematical fields, including number theory, dynamical systems, and the study of chaotic behavior. Despite vast numerical evidence and partial results, a general proof remains elusive. In this work,we propose a novel approach the Hidden Order method which reveals many patterns of the Collatz sequences and, more importantly, a singularity that will radically change the understanding of the Collatz dynamic.
Category: Number Theory
[3530] viXra:2508.0112 [pdf] submitted on 2025-08-18 20:41:10
Authors: Giovanni Di Savino
Comments: 371 Pages. (Note by viXra Admin: Further regurgitation will not be accepted; please submit article written with AI assistance to ai.viXra.org)
Natural numbers are infinite and are either prime numbers divisible by 1 and itself or composite numbers divisible by 1 and more numbers. For every number, there is a subsequent number, which is either a prime number or a composite number. Among the infinite composite numbers are the even and odd perfect numbers, which are generated by prime numbers, which are the sum of proportional numbers. The even perfect numbers are generated by the only even prime. They are the Mersenne primes, which are the result of (2^n_prime-1)*2^(n_prime-1) and, in the binary system, are the sum of the values of consecutive 1 signs. the infinite odd perfect numbers are generated by prime numbers that are the sum of numbers in proportion to one of the infinite odd prime numbers and are the result of an n_prime≥3*n_odd prime^(n-1) which is the prime number that defines the proportion of a numerical system: the 3rd, the 5th, the 7th or a system of one of the infinite odd prime numbers; the prime numbers that generate the odd perfect numbers are the sum of the value of the consecutive 1 signs of the numerical system of prime numbers ≥3.
Category: Number Theory
[3529] viXra:2508.0104 [pdf] submitted on 2025-08-16 21:16:00
Authors: Zhiyang Zhang
Comments: 12 Pages. (Note by viXra Admin: Please cite and list scientific references)
In the process of searching for counterexamples to the Riemann hypothesis, I unexpectedly proved it. Contrary to what modern mathematicians believe, although I did not create a new tool, I achieved this by constructing a sophisticated structure.
Category: Number Theory
[3528] viXra:2508.0102 [pdf] submitted on 2025-08-16 20:54:00
Authors: Kuan Peng
Comments: 9 Pages. (Note by viXra Admin: Please cite and list scientific references)
The scatter plot of Pythagorean triples exhibits distinct parabolic patterns whose origins have not been fully characterized. In this work, we derive explicit parabolic functions directly from the Pythagorean equation and demonstrate their correspondence with these patterns. The analysis shows that basic Pythagorean triples are regularly distributed on the (X,Y) plane, occurring precisely at the intersections of horizontal and vertical parabolas. The derived functions align closely with the observed parabolic structures, and a density analysis of the parabolas explains the prominence of these patterns in the scatter plot of all Pythagorean triples.
Category: Number Theory
[3527] viXra:2508.0101 [pdf] submitted on 2025-08-15 20:14:07
Authors: Fahd Alawad
Comments: 30 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This study introduces a temporal-angular model in which prime numbers are mapped onto a circular time framework, leveraging angular positions derived from modulo operations with respect to a 12-hour or 24-hour clock. The model reveals distinctive patterns of symmetry, clustering, and periodicity in the distribution of primes, suggesting that their apparent irregularity in the linear domain may transform into structured behavior within a cyclic representation of time.By analyzing the angular distribution of primes, a potential connection to the Riemann Hypothesis emerges: the observed symmetry may correspond to the regularity implied by the nontrivial zeros of the Riemann zeta function lying on the critical line u200b. The temporal-angular mapping could serve as a geometric analogue to the complex plane representation of the zeta function, offering an alternative perspective for visualizing and interpreting prime number distribution.The findings suggest that if the geometric symmetry of prime angular positions can be rigorously formalized and linked to the analytic properties of ζ(s), this approach may contribute to advancing the theoretical framework toward a proof—or deeper understanding—of the Riemann Hypothesis. Future work will involve refining the mathematical formulation, integrating Fourier and modular analysis, and establishing a direct correspondence between angular periodicity and the spectral interpretation of prime distribution.
Category: Number Theory
[3526] viXra:2508.0100 [pdf] submitted on 2025-08-15 20:06:16
Authors: Kohji Suzuki
Comments: 25 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We apply SING, a notion inspired by singularity (viXra:1812.0480 [v1]), to the Birch and Swinnerton-Dyer conjecture to suggest that the conjecture is related to the twin prime conjecture.
Category: Number Theory
[3525] viXra:2508.0080 [pdf] submitted on 2025-08-12 20:31:43
Authors: Hatem Fayed
Comments: 15 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this article, it is proved that for large |t|, all the non-trivial zeros of the Riemann zeta function must lie on the critical line, as per Riemann hypothesis.
Category: Number Theory
[3524] viXra:2508.0077 [pdf] submitted on 2025-08-12 20:37:59
Authors: Ammar Hamdous
Comments: 25 Pages. Creative Commons Attribution 4.0 International (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Several attempts have been made to generalize the Collatz sequences 3n+1, but many of them produced sequences that lack the essential structural properties of the original Collatz dynamics. Among these, the most promising known generalization is the one proposed in 2022 by Naouel Boulkaboul [2], which takes the form 3n+3^k and leads sequences to converge toward 3^k. In this work, we propose a new methodological generalization of the Collatz sequences based on a two-part transformation (1 + 2^k)n + S_k(n) if n mod 2^k ̸= 0, and n/2^k ifn mod 2^k = 0, where S_k(n) is a correction function preserving the generalized singularity previously revealed in [1]. This revised formulation ensures that all rank-1 branch beginnings exhibit the generalized singularity in binary form.Ironically, I correctly formulated the generalization of the Collatz sequences on April 4, 2025 using the auxiliary function n/2^k, but I eventually made the mistake of modifying it by n/2, given its power to prevent the rapid growth of the generalized Collatz sequences.
Category: Number Theory
[3523] viXra:2508.0067 [pdf] submitted on 2025-08-09 06:43:05
Authors: John Yuk Ching Ting
Comments: 47 Pages. Definitive proofs for Riemann hypothesis, BSD conjecture, and Polignac's and Twin prime conjectures.
Whereby all the infinitely-many prime numbers are classified as [well-defined] Incompletely Predictable entities, so must all the infinitely-many nontrivial zeros be classified as such. We outline interesting observations and conjectures about distribution of nontrivial zeros in L-functions; and [optional] use of Sign normalization when computing Hardy Z-function, including their relationship to Analytic rank and Symmetry type of L-functions. When Sign normalization is applied to eligible L-functions, we posit its dependency on even-versus-odd Analytic ranks, degree of L-function, and particular gamma factor present in the functional equations for Genus 1 elliptic curves and higher Genus curves. By carefully applying inclusion-exclusion principle, our mathematical arguments are postulated to satisfy Generalized Riemann hypothesis, and Generalized Birch and Swinnerton-Dyer conjecture. We explicitly mention underlying proven / unproven hypotheses or conjectures. We provide Algebraic-Transcendental proof (Proof by induction) as supplementary material for open problem in Number theory of Riemann hypothesis whereby it is proposed all nontrivial zeros of Riemann zeta function are located on its Critical line.
Category: Number Theory
[3522] viXra:2508.0065 [pdf] submitted on 2025-08-10 00:52:39
Authors: Christoph Bödewig
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
In this paper, we provide a complete induction proof for the following explicitformula of the Collatz iteration, parameterized by a parity sequence delta_j and then show thatthis representation is surjective, i.e., it reaches all positive integers mathbb{N}
Category: Number Theory
[3521] viXra:2508.0059 [pdf] submitted on 2025-08-09 03:17:18
Authors: Horacio Useche
Comments: 30 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
The Riemann conjeture is false. The zeros of the function $zeta(s)$ are place on $0.43leq Re(s) < 1$ interval. The straight lines with possible infinite zeros are $Re(z)=0.43$, $Re(z)=0.47$, $Re(z)=0.55$, $Re(z)=0.67$, $Re(z)=0.79$, $Re(z)=0.84$, $Re(z)=0.90$, $Re(z)=0.91$, $Re(z)=0.92$, $Re(z)=0.93$, $Re(z)=0.94$, $Re(z)=0.95$, $Re(z)=0.96$, $Re(z)=0.97$, $Re(z)=0.98$, y $Re(z)=0.99$, there are other lines with many zeros, though with minor density.We provide necessary and sufficient conditions to yields the convergence of the zeros of the Riemann zeta ($zeta$) function. A new expression for the Riemann zeta function is also deduced, in terms of a serie of sines and cosines, as expected! In the same way, we confirm the existence of the textbf{zeros by reflection} predicted by the functional equation of the zeta function and we define the concept of textbf{twin zeros} by analogy with the twin primes of the numbers theory.
Category: Number Theory
[3520] viXra:2508.0057 [pdf] submitted on 2025-08-09 02:38:09
Authors: John Yuk Ching Ting
Comments: 36 Pages. Definitive proofs for Riemann hypothesis, Polignac's and Twin prime conjectures [including proof on BSD conjecture].
Utilizing Input-White Box-Output (I-WB-O) Modeling, we outline novel applications of Mathematics for Incompletely Predictable Problems (MIPP) to unique sets and subsets from prime and composite numbers, nontrivial zeros of Riemann zeta function, and relevant number sequences from On-Line Encyclopedia of Integer Sequences (OEIS). We show MIPP is valid for selected mathematical functions, equations or algorithms that contain an equality relationship between two expressions. When applied to OEIS number sequence A228186, MIPP is also valid for an inequality. Inclusion-exclusion (I-E) principle from combinatorics removes contributions from over-counted elements in sets and subsets. By invoking I-E principle, arising consequences from MIPP formulations containing I-WB-O Models will provide necessary mathematical arguments for rigorously solving open problems Riemann hypothesis, Polignac's and Twin prime conjectures.
Category: Number Theory
[3519] viXra:2508.0055 [pdf] submitted on 2025-08-07 11:18:24
Authors: Igor Hrnčić
Comments: 3 Pages.
This paper demonstrates that many results about the Riemann Zeta Function in the literature are wrong, claiming an asymptotic estimate without checking the limit, or interchanging integration and summation when the infinite series diverges absolutely. The new Lemma is proved, proving new estimates about Zeta. These contradict known results, conditional on the truth of the Riemann Hypothesis. A stronger result holds too: any open vertical strip that holds the boundary of the Critical Strip also holds a zero of Zeta. The new Lemma can be applied to L-Functions too.
Category: Number Theory
[3518] viXra:2508.0048 [pdf] submitted on 2025-08-07 21:02:58
Authors: Edgar Valdebenito
Comments: 3 Pages.
In this note, we study an identity obtained by R. Sprugnoli in 2006.
Category: Number Theory
[3517] viXra:2508.0042 [pdf] submitted on 2025-08-06 21:02:43
Authors: Kohei Okawa
Comments: 34 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, I have repeatedly gone through various trial and error attempts, mainly inpursuit of the infinity of twin primes. Below, I have marked the failed attempts in the firsthalf, or clearly stated like that unfortunately, this is incorrect," "unfortunately, this is weakness," and have left the arguments that are clearly theoretically consistent as they are. I by no means want to write a fraudulent paper. I strongly hope that this paper will remain asource of research for future generations.
Category: Number Theory
[3516] viXra:2508.0040 [pdf] submitted on 2025-08-06 07:25:49
Authors: Theophilus Agama
Comments: 6 Pages. This paper presents a new idea, connecting existing chains to closed addition chains.
We show that Brauer and a certain class of Hansen chains satisfy the requirements for an addition chain to be closed. This puts these types of addition chain as a subfamily of the so-called closed addition chains.
Category: Number Theory
[3515] viXra:2508.0034 [pdf] submitted on 2025-08-06 20:42:13
Authors: Gustavo García, Oscar Melchor
Comments: 2 Pages. License: CC BY-NC-SA 4.0 (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
Anelementary proof of the strong form of the Goldbach Conjecture is presented: every even number greater than 2 can be expressed as the sum of two prime numbers. The strategy is based on analyzing the possible pairs of odd numbers that sum up to 2n and applying a sieve based on the divisibility of primes less than or equal to √2n. It is shown that for n ≥ 8, at least one of these pairs consists of two prime numbers.
Category: Number Theory
[3514] viXra:2508.0027 [pdf] submitted on 2025-08-05 20:38:56
Authors: Mi Zhou
Comments: 5 Pages.
This paper investigates the non-existence of positive integer solutions for equationsrelated to Fermat's Last Theorem, Beal Conjecture, and Catalan's Conjecture. For (4n1 + 1)n+(4n2 + 1)n=(4n3)n , expanding the left-hand side yields a term of the form 4nu2032+2, while the right-hand side is 4nu2032u2032, demonstrating the equation's invalidity. Fermat's Last Theorem (xn + y n=z n with n > 2) was proven by Wiles using highly complex methods. The generalized Fermat equation (x p + y q=z r) extends this, with Beal Conjecture positing no positive integer solutions when x, y, and z arecoprime—a problem yet unresolved. Catalan's Conjecture (A m=B n + 1) asserts no solutions exist beyond 3 2=2 3 + 1, proven by Preda Mihăilescu through intricate means. This study employs concise modular arithmetic to address all three conjectures.
Category: Number Theory
[3513] viXra:2508.0018 [pdf] submitted on 2025-08-04 09:05:00
Authors: V. Barbera
Comments: 7 Pages.
This paper presents an analysis of the stopping time of the 3n+1 problem based on the residue class of n.
Category: Number Theory
[3512] viXra:2508.0016 [pdf] submitted on 2025-08-04 20:12:05
Authors: Mustapha Kharmoudi
Comments: 22 Pages. (Note by viXra Admin: An abstract is required in the article)
After publishing here a summary that was widely disseminated across certain social networks, I received numerous messages requesting clarification of several of my assertions. Hence this comprehensive and detailed article.
Category: Number Theory
[3511] viXra:2508.0005 [pdf] submitted on 2025-08-03 03:34:26
Authors: Bahbouhi Bouchaib
Comments: 8 Pages. A review article
This essay addresses the often-debated question of whether an independent mathematician—one outside the framework of traditional academic institutions—can resolve one of the most enduring open problems in number theory: the strong Goldbach Conjecture. We examine historical, institutional, and methodological considerations and confront assumptions held within the mathematical community. The analysis suggests that while the challenge is formidable, advances in computational tools, access to academic literature, and creative heuristics have opened new opportunities, regardless of one's affiliation. But the hardest point in Goldbach's strong conjecture is to prove it to infinity. Even if a new theorem demonstrates Goldbach's strong conjecture, there must be no counterexample to infinity. Till now, none can map prime numbers to infinity without using primeness tests that become probabilistic when numbers tend to infinity.
Category: Number Theory
[3510] viXra:2507.0190 [pdf] submitted on 2025-07-26 20:47:46
Authors: Marcin Barylski
Comments: 4 Pages.
There are several interesting ways to depict distribution of primes like Ulam Spiral, Klauber Triangle or the Sacks Number Spiral. In all cases, Prime Number Theorem describes the asymptotic distribution of such numbers among the positive integers. This work is devoted to illustration of primes of form p x q +- C in a way that allows to search for clusters (so called islands of primes). The direct goal of this experimental work is to locate islands with the largest surface area and potentially discover some further patterns in distribution of primes.
Category: Number Theory
[3509] viXra:2507.0186 [pdf] submitted on 2025-07-26 01:11:14
Authors: Bo Zhang
Comments: 4 Pages.
The Riemann hypothesis is proved true through a linear combination of the real and imaginary parts of the Riemann ξ-function.
Category: Number Theory
[3508] viXra:2507.0155 [pdf] submitted on 2025-07-21 01:55:50
Authors: Theophilus Agama
Comments: 12 Pages.
We prove that a certain class of infinite sequences whose finite truncation is an addition chain must have a zero logarithmic density. This result is generic and can be applied to particular known infinite sequences with this property.
Category: Number Theory
[3507] viXra:2507.0153 [pdf] submitted on 2025-07-21 21:18:33
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper Riemann rearrangement theorem is going to be analyzed on a single example and it is going to be explained that the proof of the theorem is incomplete and wrong, That means that it does not matter how you rearrange the elements of the series, the sum would always stay the same. The reason that "rearranged" series does not have the same sum as the original series, is in the hidden omission of infinite number of elements that are contained in the original series. The content is presented in the form of explanation of a magic trick (since the claim of the theorem sounds as a real magic).
Category: Number Theory
[3506] viXra:2507.0143 [pdf] submitted on 2025-07-20 20:23:24
Authors: Fawang Su
Comments: 13 Pages.
By setting the even - exponent orbit of the 3x + 1 iteration to construct and solve the initial number, and let the odd - orbit of this initial number iterate under the control of the even - exponent orbit. Through this construction method, it is proved that the 3x + 1 iteration step length of a specific type of number can be arbitrary or even infinite.
Category: Number Theory
[3505] viXra:2507.0135 [pdf] submitted on 2025-07-19 22:48:56
Authors: Kuan Peng
Comments: 21 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
Pythagorean triples are generated with Euclid’s formula. But how this formula was derived by or before Euclid is a mystery. We have derived Euclid’s formula directly from Pythagorean equation and classified all Pythagorean triples in a 3D table. The equation X^2+Y^j=Z^2 is proven to have infinitely many integer solutions. By comparing Pythagorean equation with Fermat’s equation for n=3 we were able to explain why Fermat’s equation with n=2 has integer solutions while with n [equal or larger than] 3 it has not. We propose an algebraic method to work Fermat’s last theorem
Category: Number Theory
[3504] viXra:2507.0134 [pdf] submitted on 2025-07-19 01:35:53
Authors: Izzie Boxen
Comments: 63 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems and some extant conjectures being proved. Included are a new and independent proof for Bertrand’s postulate, proofs for Lim((p_[n+1]-p_n)/p_n)=0 , and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann.
Category: Number Theory
[3503] viXra:2507.0116 [pdf] submitted on 2025-07-16 15:54:56
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper, we formulate an optimization problem for addition chains.
Category: Number Theory
[3502] viXra:2507.0110 [pdf] submitted on 2025-07-15 08:44:04
Authors: Simon Plouffe
Comments: 8 Pages.
A series of large-scale tests were performed on the first 1000 billion digits of the numberπ. First a direct visual test as well as a test using thousands of sequences from the OEIS catalog.The purpose of these tests is to detect possible patterns. Other tests were made on tenmathematical constants to 1 billion decimal places.
Category: Number Theory
[3501] viXra:2507.0103 [pdf] submitted on 2025-07-14 11:05:05
Authors: Muhammad Razzaq Aman Wattoo, Mehar Ali Malik, Iqra Aftab
Comments: 6 Pages.
In this paper, we have developed a set S named as Aman’s set that is union of collection of sets generated by arithmetic sequence. Then we have used this set to generate two separate complete lists of prime numbers and composite odd numbers. Also we have run this set to find the finite list of the prime numbers and composite numbers in python successfully.
Category: Number Theory
[3500] viXra:2507.0102 [pdf] submitted on 2025-07-14 20:41:24
Authors: Younghwan Yun
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We propose a conservative structural framework to address the twin prime conjecture, aiming to demonstrate the unavoidable recurrence of twin prime pairs across the number line. By systematically applying the inclusion-exclusion principle within bounded intervals [p2 n−1, p2n), where pn is the n-th prime, we estimate the minimal lower bound of surviving 6k ± 1 pairs after sieving out all multiples of smaller primes. Our analysis shows that any composite survivor within such intervals would require a prime factor at least as large as pn, leading to a contradiction by exceeding the interval’s up-per bound. We derive an explicit minimal estimate Tn for the number of twin primepairs and show that it grows unboundedly with n. This non-probabilistic approach provides a concrete methodological pathway suggesting that the periodic sieve structure necessarily sustains infinitely many twin prime pairs, offering strong structural support for the twin prime conjecture.
Category: Number Theory
[3499] viXra:2507.0092 [pdf] submitted on 2025-07-14 01:24:22
Authors: Hassan Bouamoud
Comments: 3 Pages.
This is a tentative [proof of] Andric's conjecture using a known result of Baker-Herman-Bintz.
Category: Number Theory
[3498] viXra:2507.0066 [pdf] submitted on 2025-07-09 23:08:01
Authors: Theophilus Agama
Comments: 4 Pages.
Let ��u2099: su2080 = 1 < su2081 = 2 < ⋯ < su2095 = nbe an addition chain leading to n. Define the normalized profile vu2099(x) := s_{⌊xh⌋} / nfor any x ∈ [0,u202f1] and set xᵢ := i / h. We show that for any fixed x ∈ [0,u202f1] there exists an index i with 0 ≤ i ≤ h such that x·hu2044n ≤ vu2099(xᵢ) ≤ x + 1u2044h = x + o(1). This implies that no matter how an addition chain is built, at each fraction x ∈ [0,u202f1], there is some term whose normalized size is in the interval [x·hu2044n, x + o(1)]. This may be viewed as a Bertrand-type result in addition chains.
Category: Number Theory
[3497] viXra:2507.0049 [pdf] submitted on 2025-07-06 21:08:56
Authors: Denis Micheal Odwar
Comments: 9 Pages.
For m ∈ Z, let N = 2m ≥ 8 and GN be a set of goldbach primes, p, of N defined as GN = {p : p ≤ N/2 andp ∤ N}. By denoting the cardinality of GN by | GN | or g(N), we show that ∀N, | GN |> 0, and the set of these cardinalities, {| GN |}, is equal to the set of natural numbers i.e {| GN |} = N. We finally prove the famous Goldbach binary |GN| conjecture by showing that i=1 µ(bi)Λ(bi) < 0, whenever bi = N −pi with pi ∈ GN,i ∈ N and 1 ≤i≤| GN | . In particular we show that every N is a sum of two distinct primes.
Category: Number Theory
[3496] viXra:2507.0029 [pdf] submitted on 2025-07-04 16:45:11
Authors: Walter A. Kehowski
Comments: 6 Pages.
The minimal set of primes in base b is a finite set of primes with the following property: if p is a prime, then there exists at least one element q of the minimal set such that the digits of q in base b form a subsequence of the digits of p in base b. The purpose of this note is construct the minimal set of primes in base 12.
Category: Number Theory
[3495] viXra:2507.0028 [pdf] submitted on 2025-07-04 16:45:44
Authors: Theophilus Agama
Comments: 5 Pages.
We prove that the spacing between consecutive termsin an addition chain with non-decreasing τ -track can be generated by adding two previous terms in the chain.
Category: Number Theory
[3494] viXra:2507.0020 [pdf] submitted on 2025-07-03 17:36:12
Authors: Chloe Williams
Comments: 8 Pages.
In this paper, I will prove all positive odd integers link incrementally through connected orbits to a value 2^c where c ∈ N. These orbits connect because of a function I defined based on the connection between all odd and 8mod12 positive integers. Applying this function infinitely many times to all positive odd integers always leads to a connection to power of 2 orbits. With these results, I’m able to prove the conjecture.
Category: Number Theory
[3493] viXra:2507.0011 [pdf] submitted on 2025-07-02 19:49:00
Authors: Bo Zhang
Comments: 10 Pages.
Goldbach's conjecture is proved through the new understanding of the fundamental theorem of algebra.
Category: Number Theory
[3492] viXra:2507.0010 [pdf] submitted on 2025-07-02 23:44:41
Authors: Bahbouhi Bouchaib
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This article presents a novel computational method for decomposing even numbers into sums of two prime numbers, a problem known as Goldbach’s Conjecture. Using optimized heuristics based on modular arithmetic and probabilistic constraints, I have developed a publicly accessible website capable of processing even numbers up to 1018 which is a historic record. My method, grounded in decades of theoretical insights and refined by computational efficiency, offers a new way to visualize and explore prime pair decomposition and contributes to the ongoing exploration of one of number theory’s most famous conjectures. The latest successful version of my new website for Goldbach's decomposition is available at: https://b43797.github.io/Bahbouhi-decomposing-Goldbach-conjecture2025/.
Category: Number Theory
[3491] viXra:2506.0143 [pdf] submitted on 2025-06-25 15:24:01
Authors: Theophilus Agama
Comments: 13 Pages.
We develop explicit bounds for the gap between consecutive terms in an addition chain leading to a fixed target 2^m<n<2^{m+1}.
Category: Number Theory
[3490] viXra:2506.0124 [pdf] submitted on 2025-06-21 21:23:18
Authors: Giovanni Di Savino
Comments: 1 Page. (Note by viXra Admin: Please cite and list scientific references)
Fermat, studying Book II of Diophantus' Arithmetica, in the pages dedicated to the problems and observations around the Pythagorean Theorem, in a marginal note of the book, reports: "It is impossible to write a cube as the sum of two cubes or a fourth power as the sum of two fourth powers or, in general, no number that is a power greater than two can be written as the sum of two powers of the same value". This conjecture, known for many years as Fermat's last theorem, became a theorem because Prof. Andrew Wiles demonstrated that c^n≥3 ≠ a^n≥3 + b^n≥3
Category: Number Theory
[3489] viXra:2506.0121 [pdf] submitted on 2025-06-20 04:59:16
Authors: Jose Risomar Sousa
Comments: 11 Pages.
A major breakthrough introduced in this paper is a generalization of the Riemann functional equation that has a broader validity domain than the existing one from the literature. The insight that led to this new relation came from a new formula for the zeta function created herein that implies the Riemann functional equation. Further developments that stem from new formulae introduced previously are also discussed.
Category: Number Theory
[3488] viXra:2506.0112 [pdf] submitted on 2025-06-20 20:12:44
Authors: Theophilus Agama
Comments: 6 Pages.
Let us define the function F(m, r) as the number of integers n in the interval [2^m, 2^{m+1}) such that ι(n) ≤ floor(m + r), where r = c·m / log m for some constant c satisfying 0 larger than c smaller than log 2. Next, define α = c + log 2 − (1/4)(1 − o(1)). By applying the Chernoff inequality from probability theory and using ideas from De et al. (2025), we obtain the following improved upper bound: F(m, c·m / log m) ≤ exp[ α·m − (1 − ε)·(c·m·log log m) / log m ] for any small ε > 0 as m tends to infinity.
Category: Number Theory
[3487] viXra:2506.0110 [pdf] submitted on 2025-06-20 20:10:04
Authors: Theophilus Agama
Comments: 6 Pages.
Define the function F(m, β(m) − m) as the number of integers n in the interval [2^m, 2^{m+1}) such that l(n) ≤ β(m). By applying ideas from De Koninck et al. (2025), we obtain the following general upper bound: F(m, β(m) − m) ≤ exp[ (β(m) − m) · (2·log β(m) + (1 + ε)·2·log log m + O(1)) ] for any small ε > 0 as m tends to infinity. This result generalizes a recent bound proved in the work of De Koninck and collaborators.
Category: Number Theory
[3486] viXra:2506.0109 [pdf] submitted on 2025-06-19 20:58:23
Authors: Yibing Xiong
Comments: 84 Pages. (Note by viXra Admin: Please cite and list scientific references)
Disordered events can be divided into two types: disordered deterministic events - generalized events, disordered random events - random events - narrow events; Establishing probability axioms, boundary axioms, etc., proposing four important original works: probability method, boundary, subdivision probability, Xiong's sieve; Establish General Probability Theory. Using analytical methods to solve mathematical problems such as the prime number theorem, Riemann hypothesis, twin prime numbers, densest K-tuple prime numbers, K-tuple prime numbers, Goldbach's conjecture, Gaussian lattice points, etc.
Category: Number Theory
[3485] viXra:2506.0096 [pdf] submitted on 2025-06-18 19:43:06
Authors: Theophilus Agama
Comments: 9 Pages. (Note by viXra Admin: Please don't use LaTeX codes in the abstract!)
[This note provides a quantitative bound for the number of primes in an addition chain]
Category: Number Theory
[3484] viXra:2506.0080 [pdf] submitted on 2025-06-16 01:24:22
Authors: Ahmed Souissi
Comments: 3 Pages.
Abstract. While exploring Dirichlet L-functions as part of a final-year project, I stumbledupon a surprising property: a wave function ψp(x) = χ(p)eiγx, where χis a non-trivial Dirichlet character modulo q, γ is a non-trivial zero of L(s,χ), and p is a prime, produces a discrete Fourier transform ψp(k) with a dominant peak at k ≡ p−1 mod q. This "mirror symmetry"suggests a deep arithmetic structure linking primes to their modular inverses. I formalize this observation with a quantitative conjecture, provide numerical evidence for q = 5,13,17, andoffer a partial theoretical analysis using Gauss and Kloosterman sums. Potential applicationsin quantum physics and cryptography are discussed.
Category: Number Theory
[3483] viXra:2506.0079 [pdf] submitted on 2025-06-15 22:29:52
Authors: Walter A. Kehowski
Comments: 8 Pages.
A sublime number is a number such that both its number of divisors and sum of divisors are both perfect numbers. The number 12 is the first sublime number. This paper gives Kevin Brown's construction of even sublime numbers a modern mathematical development.
Category: Number Theory
[3482] viXra:2506.0055 [pdf] submitted on 2025-06-12 22:24:39
Authors: Taha M. Muhammad
Comments: 4 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
a, b, c, d, e, f, g ∈ N_+ ⟺ Euler Perfect Box Let a, b, c, d, e, f, r, k ∈N_+. I have to prove that g ∈N_+ Let a < b < c, & [(a
Category: Number Theory
[3481] viXra:2506.0054 [pdf] submitted on 2025-06-12 22:24:29
Authors: Taha M. Muhammad
Comments: 10 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
1- Let x ∈ Loop of Collatz Sequence = LS(n), n, x,y,z,t,u,v ∈ N+ i) I proved LS(n) = { }, {x}, {x,y}, {x,y,z,t}, {x,y,z,t,u,u2026,v }, and {x,y,z,t,u,u2026,v,u2026} is false. ii) I proved LS(n) = {x,y,z} = {4,2,1} = {4,2,1, 4,2,1} =u2026= {4,2,1, u2026, 4,2,1} is true ∀n ∈ N+ 2- I created special sketch to find algebra expressions equal to x. 3- I created special Table to find values of elements in the LS(n). 4- I created a pattern to find values of x ∈ LS(n). 5- Observation: let r, k ∈ N+, r = number of elements in the LS(n):i) if r = 3k ⇒ x ∈ N+ ⇒ LS(n) = {1,2,4} ∀n ∈ N+ ii) if r = 3k + h, such that h=1,or 2⇒ x ∉ N+ ⇒ LS(n) has no loop ∀n ∈ N+ ∴ LS(n) ={4,2,1},∀n ∈ N+.
Category: Number Theory
[3480] viXra:2506.0053 [pdf] submitted on 2025-06-12 22:36:30
Authors: Taha M. Muhammad
Comments: 5 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
Fermat’s Last Theorem: There are no natural numbers (1, 2, 3, u2026) a, b, and c such that a^n+b^n = c^n, in which n is a natural number greater than 2. 1-Taha^' s Logic Equations Fact a=b (True or False) & c=d (True)⇒(a-c=b-d) Fulse⇔a=b ( False) 2- Taha^' s Coefficient Fact: [a + b = c ⇒ ax + by= cz] ⇔ x = y = z3- Taha’s (N_+) & Three-Sided Geometric Shapes Fact below: N_+ = (Right Triangles) ∪(Acute Triangles) ∪(Obtuse Triangles) ∪ (Segments c=a + b) ∪ (Segments c>a + b).
Category: Number Theory
[3479] viXra:2506.0052 [pdf] submitted on 2025-06-12 23:01:21
Authors: Hajime Mashima
Comments: 79 Pages. In Japanese
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[3478] viXra:2506.0042 [pdf] submitted on 2025-06-10 22:32:14
Authors: Edgar Valdebenito
Comments: 3 Pages. (Note by viXra Admin: Further repetition may not be accepted)
We give some formulas related to the number 239.
Category: Number Theory
[3477] viXra:2506.0041 [pdf] submitted on 2025-06-09 20:49:25
Authors: Ahcene Ait Saadi
Comments: 7 Pages. (Note by viXra Admin: Please cite and list relevant scientific references)
In this article, I studied the behaviour of natural integers, after performing operations on them. The results I have obtained are intriguing and I cannot explain them. I share this original work with young researchers for deepening and improvement.Key words: Integers, operations, sequences.
Category: Number Theory
[3476] viXra:2506.0030 [pdf] submitted on 2025-06-08 19:30:38
Authors: Jean-Yves Boulay
Comments: 40 Pages.
Here is expanded the concept of Sophie Germain prime and safe prime to ultimate numbers. More, this mathematical mechanism is also applied to the set of all whole numbers (ℕ) which are differentiated into ultimate and non-ultimate numbers. Thus, a complete study of the set ℕ is undertaken. This global investigation, universally broadening the mechanical-mathematical concept of Sophie Germain, makes it possible to propose a genetics of numbers very similar to biological genetics. According to these new numeric genetic criteria, in their start top organization, geometric distribution of whole numbers in various closed matrices, is organized into perfect ratios to exact 3/2 or 1/1 value.
Category: Number Theory
[3475] viXra:2506.0024 [pdf] submitted on 2025-06-07 20:13:06
Authors: Abdelrahman M. Mohammed
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We begin by observing a striking "mirror-complement" pattern in the binary digits of √2: whenever, at any position, a run of k equal bits is separated by a single opposite bit from another run of bits, those two runs must have equal length. Restricting to prime-indexed positions, the same pattern remains perfectly true for millions of primes. This phenomenon is a direct consequence of the classical digit-by-digit square-root algorithm in base 2, because each comparison uses4Pn + 1 = 2 (2Pn) + 1,i.e. "copy + complement + copy."From this insight we build a two-rectangle coding on T² whose itinerary reproduces the binary digits of √2. A measurable conjugacy to the (½,½) Bernoulli shift allows us to apply Chung—Smorodinsky’s bounded-coboundary theorem (1967), showing each cylinder-indicator has a uniform sup-norm bound. Telescoping that coboundary yields a universal O(Nu207b¹) discrepancy bound on every length-ℓ binary block, proving base-2 normality of √2. Finally, van der Corput differencing and Wall’s criterion transfer the same O(Nu207b¹) bound to every integer base B ≥ 2, establishing that √2 is normal in all bases.This paper unifies these ideas—starting from the prime-indexed mirror pattern and culminating in a gap-free, self-contained proof of full normality of √2.
Category: Number Theory
[3474] viXra:2506.0023 [pdf] submitted on 2025-06-07 20:07:13
Authors: Viktor Arvidsson
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We prove the twin prime conjecture, asserting infinitely many integers x such that both x and x + 2 are prime. Construct an N × N diagonal matrix HN with diagonal entries Dx = 0 if x, x + 2 prime, and Dx ≥ 3 10 otherwise. The kernel dimension equals the number of twin primes up to N, denoted |TN|. A heat-trace argument on Tr[exp(−2HN)] forces |TN| → ∞, proving the theorem. We refine parameter bounds, standardize notation, and add explicit estimates for key constants.
Category: Number Theory
[3473] viXra:2506.0019 [pdf] submitted on 2025-06-06 19:51:59
Authors: Hans Hermann Otto
Comments: 5 Pages.
This is only a first short essay about the ‘angel’ number 123, a symbol for balance, human creativity, divine order or trinity of God the Father, the Son and the Holy Spirit. The author thinks that important information about this number is not well known. Here this enigmatic number is interpreted mathematically as beautiful golden relation, connected with the golden mean and its fifth power that is vice versa governed by phase transitions from particle up to cosmic scale.
Category: Number Theory
[3472] viXra:2506.0018 [pdf] submitted on 2025-06-06 19:47:59
Authors: Fabrice Trifaro
Comments: 45 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Using a comprehensive approach, this paper aims to demonstrate, clearly and rigorously, the validity of the Collatz conjecture. To this end, the original 3n+1 iteration is reformulated by isolating the odd terms into sequences referred to as R-Cz sequences. These sequences are analyzed through their structural properties and their distribution among the odd natural numbers. As a first essential result, it is shown that they do not admit non-trivial cycles: the only possible cycle is the trivial one, of value and length 1. Two independent proofs that all R-Cz sequences converge are then presented. The first, combinatorial in nature, relies on the finiteness of intervals that could possibly separate terms of the sequences. The second, set-theoretic, is based on a contradiction between the countability of the odd integers and the uncountable cardinality of the hypothetical divergent R-Cz sequences. Both methods lead to the same conclusion: all Collatz sequences eventually enter the cycle (1,4,2).
Category: Number Theory
[3471] viXra:2506.0009 [pdf] submitted on 2025-06-03 23:39:40
Authors: Predrag Terzić
Comments: 2 Pages. (Note by viXra.org Admin: Please cite and list sceintific references)
We present new continued fraction representation of constant pi.
Category: Number Theory
[3470] viXra:2506.0006 [pdf] submitted on 2025-06-02 19:37:53
Authors: Predrag Terzić
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
We present a new, specific primality test for numbers of the form N = 4p^n - 1, where p is an odd prime and n > 0. The test is a generalization of the Lucas-Lehmer test for Mersenne numbers and relies on a sequence defined by Dickson polynomials. We prove that, under a certain condition, N is prime if and only if the n-th term of a specific sequence is congruent to zero modulo N. This provides a deterministic primality test for this family of numbers.
Category: Number Theory
[3469] viXra:2506.0001 [pdf] submitted on 2025-06-01 20:56:09
Authors: Mar Detic
Comments: 4 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
This document introduces and investigates a criterion for defining "primality" within the sequence of odd integers Ak = 2k + 1. The criterion is based on the greatest common divisor (GCD) between a term Ak and the preceding partial sums Sj = j2 + 2j of the same sequence. We formally define the sequence, its partial sums, and the proposed primalitycriterion. A proof is provided demonstrating that all standard prime numbers within the sequence are classified as "prime" by this new definition. Furthermore, computational observations suggest that all composite numbers in the sequence are classified as "not prime," including Carmichael numbers, which are known for their pseudoprime properties. This leadsto a conjecture that the proposed criterion is equivalent to the standard definition of primality for terms in this specific arithmetic progression.
Category: Number Theory
[3468] viXra:2505.0206 [pdf] submitted on 2025-05-31 20:18:48
Authors: YoungHwan Yun
Comments: 7 Pages.
We introduce a structural decomposition framework for perfect numbers, based on additive and multiplicative symmetries among their proper divisors. Under this model, we derive a system of proportional identities that all known even perfect numbers satisfy. By analyzing the integer conditions required for recursive consistency in the divisor sequences, we prove that such a structure necessarily implies the presence of the even divisor 2, thereby excluding the possibility of an odd perfect number conforming to this model. Although primarily developed for perfect numbers, the additive portion of the framework may also encompass semiperfect numbers, which share similar but relaxed divisor-sum properties. Our results support the long-standing conjecture that no odd perfect numbers exist and suggest a broader structure for divisor-based classification of integers.
Category: Number Theory
[3467] viXra:2505.0205 [pdf] submitted on 2025-05-31 20:15:59
Authors: Younghwan Yun
Comments: 14 Pages.
This paper presents a symmetry-based approach to the Generalized Riemann Hypothesis, focusing on the structure of nontrivial zeros of the completed Dirichlet L-function. By examining the relationship between the functional equation and complex conjugation, the argument shows that each nontrivial zero implies the existence of a symmetrically paired zero. This pairing, when interpreted through a restricted application of the Schwarz Reflection Principle at the zero points, leads to the conclusion that all nontrivial zeros must lie on the critical line. The analysis is further extended to generalized cases, including product forms, which consistently reduce to the same critical line condition. This work therefore proposes a comprehensive and logically consistent framework that supports the truth of the Generalized Riemann Hypothesis.
Category: Number Theory
[3466] viXra:2505.0204 [pdf] submitted on 2025-05-31 20:13:58
Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. (Note by viXra Admin: Further repetition will not be accepted!)
In this paper, we assume that the explicit abc conjecture of Alan Baker (2004) is true, we give the proof that c smaller than R² is true, it is one of the keys to resolve the mystery of the abc conjecture. Some numerical examples are given.
Category: Number Theory
[3465] viXra:2505.0186 [pdf] submitted on 2025-05-28 00:30:50
Authors: Junho Eom
Comments: 14 Pages.
This paper identified the characteristics of prime numbers within a limited boundary, defined primes between quadratic intervals, and generalized Legendre’s conjecture. Regarding the boundary, every integer less than m was defined as the 1st boundary and it expanded to the mth boundary within m^2. Thus, each boundary contained m elements. Except for 1, every integer produced a sine wave from the 1st boundary; as a result, only prime waves affected the remaining boundaries from the 2nd to mth by generating the composites and new primes (Series I). Therefore, the number of new primes in each boundary could not exceed ��(1st boundary) or ��(m), where ��(x) was the number of primes less than or equal to x, and it enabled the estimation of the total number of primes within m^2 (Series II). Based on Series I and II, the quadratic intervals between ��(m^2) and ��((m + 1)^2) were identical to the sum of the last two boundaries, expressed as 2·βm+1·��(m), where βm+1 was the ratio of ��((m + 1)^2) to ��(m + 1)·(m + 1) (Series III). This led to the conclusion that Legendre’s conjecture satisfied while
0.8986·��(P) < 2·βP·��(P) < ��(P) (prime P > 113), or 2·βm+1·��(m) ≤ 2·��(m) (integer m ≥ 2).
Category: Number Theory
[3464] viXra:2505.0167 [pdf] submitted on 2025-05-25 03:13:51
Authors: Abdelmajid Ben Hadj Salem
Comments: 141 Pages. Comments welcome
In this booklet, I present my proofs of open conjectures on the theory of numbers.It concerns the following conjectures:- The Riemann Hypothesis.- Beal's conjecture.- The conjecture c<rad^{1.63}(abc).- The explicit abc conjecture of Alan Baker.- Two proofs of the abc conjecture.- The conjecture c<rad^2(abc).
Category: Number Theory
[3463] viXra:2505.0165 [pdf] submitted on 2025-05-23 20:12:06
Authors: David Adam, Laurent Denis
Comments: 21 Pages. In French
n 1995, Thakur introduced analogues of hypergeometric functions for Fq [T ]. In this article, we show that the exceptional set of such a function is trivial. This confirms a conjecture stated by Thakur and his co-authors in 2008. Furthermore, we prove a Lindemann-Weierstrass-type theorem for an uncountable class of functions containing hypergeometric functions. The method used allows us to obtain the first transcendence measures (of quality comparable to that of characteristic 0) for values u200bu200bat algebraic arguments of these functions. Finally, in the last section, we exhibit the first infinite families of hypergeometric functions whose exceptional set can be shown to be trivial in the P-adic domain.
Category: Number Theory
[3462] viXra:2505.0158 [pdf] submitted on 2025-05-23 20:01:39
Authors: Dawit Geinamo
Comments: 58 Pages. 58
The objective of this study is to present rigorous proofs for Collatz conjecture and introduce some interesting behavior of the Kaakuma sequence that is a vast generalized form of Collatz sequence. We analyze the behavior of Kaakuma sequence such as scaling up, scalingdown, translation, function iteration and uniform growth of inverse tree. In addition to this we investigate relationship of increasing rate, number of iterations of cycles, gap in cycles, and densities of cycles ofthe Kaakuma sequence and evaluate consistency of tree size density after scaling. Our investigation culminates in the formulation of a set of conjectures encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz conjecture and contribute to our understanding of the underlying mathematical structure.
Category: Number Theory
[3461] viXra:2505.0150 [pdf] submitted on 2025-05-22 20:25:46
Authors: Bahbouhi Bouchaib
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references of other authors and submit article written with AI assistance to ai.viXra.org)
In this article I apply classical statistical laws to analyze prime numbers assimilated to populations. The statistical analysis focuses on prime numbers in the intervals [0 - S/2] and [S/2 — S] with S an even > 4. The results show that the even number S > 4 is enclosed by two populations of prime numbers P in the interval [0 - S/2] and Q in [S/2 - S] which have approximately the same standard deviation relative to their means. Two other subpopulations P' included in P and Q' included in Q which satisfy the strong Goldbach conjecture (P' + Q' = S) also have the same standard deviation and superimpose or overlap with a slight variation. This result shows that an even number is enclosed by two populations P' and Q' of prime numbers which are symmetric with respect to S/2 and therefore S = P' + Q'. This result also shows that any natural number N > 4 is enclosed by at least two equidistant and symmetric prime numbers. Therefore for every N > 4 there exists a number t < N such that N — t = P' and N + t = Q' are primes and so 2N = P' + Q'.
Category: Number Theory
[3460] viXra:2505.0131 [pdf] submitted on 2025-05-20 20:08:41
Authors: Mohamed Amine Boussadan
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references; Please submit article written with AI assistance to ai.viXra.org)
Abstract: Before starting to build the sequence and its topological steps, we decide the specific hypotheses that follow the pattern of prime numbers, which explain the complex gaps between prime numbers that appear random. This is done by using graphical networks (Descriptive and Euclidean Geometry). In order to study the properties that describe the distinctive nature of it, a geometric shape was transformed into a real rectangle surrounded by a circle because of no verification of the condition of a triangle which makes them straight line that grows regularly according to a vector function, whichresulted in knowing some linear relationships between variables.
Category: Number Theory
[3459] viXra:2505.0126 [pdf] submitted on 2025-05-19 21:18:57
Authors: Ahcene Ait Saadi
Comments: 8 Pages.
In this article, I demonstrate the resolution of the 5th degree equation, with algebraic radicals. I contradict Galois theory. In my demonstration, I use two invariant polynomials of degrees 8 and 6, wich I discovered*. After identification, I cancel the coefficients 7, 5 and 3 in the polynomial of degree 8 and the coefficient of degree 5 and 3, in the polynomial of degree 6. For the coefficient of degree 1, I make a combination between the polynomials of degree 8 and 6 to eliminate it. Finally I find an equation of degree 8 bisquare that I can solve. By solving the system of equations, of variables m and p, which are the coefficients of the powers 5 and 3, I’m forced to involve the equation of degree 5. That allows me to eliminate the coefficient of degree 3 from the equation of degree 8, I realise that I can find the solutions of the equation of degree 5 with a free variable. The solutions of an equation with a free variable, I have already done this with the equation of degree 2, that I discovered.
Category: Number Theory
[3458] viXra:2505.0117 [pdf] submitted on 2025-05-19 01:46:49
Authors: Trinh Tung Lam
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
The Prime Spectrum Model investigates the connection between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers using spectral analysis. A wave function is constructed from 40,000 zeros and analyzed using Fourier Transform and Short-Time Fourier Transform (STFT). Detected frequency peaks align with the natural logarithms of prime numbers, achieving RMSE = 0.0600 and Spearman correlation ≈ 1.0. The model successfully identifies the first 50 primes and extrapolates to higher ones. This work offers insights into the Riemann Hypothesis and opens applications in physics, signal processing, and complexity theory.
Category: Number Theory
[3457] viXra:2505.0110 [pdf] submitted on 2025-05-17 00:55:57
Authors: Bube Ibekwe
Comments: 6 Pages.
This paper investigates the variance of transformed twin primes, where each prime pair (p, p+2) is mapped to k = (p+1)/6. Initial analysis suggested unexpected growth patterns in the normalized variance, seemingly contradicting theoretical expectations. Through large-scale computation of twin primes up to one billion, we demonstrate that these apparent anomalies arise from normalization artifacts. The true variance follows a quadratic growth pattern, with our empirical results closely matching predicted scaling behavior. We resolve the paradox by showing how the sparse distribution of twin primes distorts normalized statistical measures. Our findings highlight critical pitfalls in analyzing prime distributions and provide new insights into the statistical behavior of twin primes.
Category: Number Theory
[3456] viXra:2505.0107 [pdf] submitted on 2025-05-15 01:13:35
Authors: Shanzhong Zou
Comments: 11 Pages.
This paper proposes a new number theory method (comb method) to prove the unending existence of prime twins. the set of natural numbers is viewed as a union of two sets that don’t intersect s-element sets and h-element sets, these elements are selected by a series of combs. By analyzing the distribution of these elements, we get our result.
Category: Number Theory
[3455] viXra:2505.0086 [pdf] submitted on 2025-05-14 19:48:09
Authors: Theophilus Agama
Comments: 6 Pages. (Note by viXra Admin: Further repetition will not be accepted)
In this note, we introduce the energy method for constructing the length of addition chains leading to $2^n-1$. This method is a generalization of the Brauer method. Using this method, we show that the conjecture is true for all addition chains with ''low'' energy.
Category: Number Theory
[3454] viXra:2505.0085 [pdf] submitted on 2025-05-14 19:46:41
Authors: Travis Shane Taylor
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We propose a symbolic gateword encoding of the Collatz transformation, demonstrating that all positive integers reduce to the fixed point 1 via finite symbolic collapse. By reformulating the Collatz function as a compressible grammar and defining collapse as a symbolic entropy-reduction process, we offer a constructive resolution to the conjecture and frame it as a computational attractor with implications for number theory, complexity, and information physics.
Category: Number Theory
[3453] viXra:2505.0075 [pdf] submitted on 2025-05-12 06:57:35
Authors: Benjamin Xavier Last
Comments: 13 Pages. bxlast@hotmail.com
This paper introduces Last Base Mathematics (LxB), a novel numerical and geometrical framework based on an alternating base system rooted in base-12 with a secondary recursive base, commonly base-5. LxB minimizes symbolic footprint by employing radial, modular divisions to represent numbers through layered circular structures, maintaining internal coherence without requiring decimal expansion. We formalize the arithmetic operations intrinsic to LxB, including native addition, subtraction, multiplication, and division rules respecting alternating bases. The system's applications are explored in the contexts of timekeeping, music theory, geometric compression, simulation, and data storage. Future research avenues include potential applications to harmonic measurement models, modular computation, and quantum phase structures. This work positions LxB as a compact, constructible alternative to linear base systems, suitable for both theoretical exploration and practical modeling.
Category: Number Theory
[3452] viXra:2505.0069 [pdf] submitted on 2025-05-11 19:37:01
Authors: Jay Y. Jeong
Comments: 4 Pages. (Note by viXra Admin: Full author name is required on the article)
We present a generalization of the well-known Lemma of Lifting the Exponent (LTE), introducing a novel valuation function. Using this framework, we outline a new approach to Fermat’s Last Theorem that relies solely on elementary number theory techniques.
Category: Number Theory
[3451] viXra:2505.0067 [pdf] submitted on 2025-05-10 09:50:55
Authors: Shanzhong Zou
Comments: 4 Pages.
This paper discovered the relationship between the relationship between One Variable Quadratic Equation and odd perfect numbers, and with the help of Veda's theorem, proved there is no odd perfect number.
Category: Number Theory
[3450] viXra:2505.0062 [pdf] submitted on 2025-05-10 20:39:54
Authors: Theophilus Agama
Comments: 13 Pages.
We prove the prime obstruction principle and the sparsity law. These two are collective assertions that there cannot be many primes in an addition chain.
Category: Number Theory
[3449] viXra:2505.0061 [pdf] submitted on 2025-05-09 21:31:24
Authors: Theophilus Agama
Comments: 7 Pages.
We introduce a new class of addition chains and show the numbers for which these chains are optimal satisfy the Scholz conjecture, precisely the inequality $$iota(2^n-1)leq n-1+iota(n).$$
Category: Number Theory
[3448] viXra:2505.0057 [pdf] submitted on 2025-05-09 21:20:56
Authors: Halaoui Ayyoub
Comments: 3 Pages.
This paper investigates the non-random digital root patterns observed in prime k-tuples (e.g., twin primes, prime triplets). By analyzing over 10u2078 primes from the Twin Prime Database and OEIS, we demonstrate a statistically significant bias toward specific digital root sequences (e.g., (8,1) for twins, (5,7,2) for triplets) with frequencies up to 3.5× higher than random expectation. We explain these patterns using modular arithmetic in Z/9Z and sieve theory, while proving that constraints on prime divisibility limit the maximum k-tuple length to 7 primes. This study bridges computational evidence with theoretical number theory, suggesting that primes exhibit quasirandom behavior with deep underlying structure.
Category: Number Theory
[3447] viXra:2505.0054 [pdf] submitted on 2025-05-08 19:47:04
Authors: Bouazad El Bachir
Comments: 297 Pages.
Riemann Hypothesis is a conjecture that states that all non trivial zeros of Riemann function are located on critical strip exactly on 1/2.This conjecture has been unsolved for over 160 years. In this proof that contains 294 pages, I will prove the conjecture of Riemann hypothesis using theorems and formulas that have never discovered before , I will also prove that there is and other function that is similar to Riemann Zeta Function and all its non trivial zeros lie exactly on critical strip — 1/2 If mathematician like Ramanujan has found the sum of this infinite series : 1+2+3+4+5+6+7+u2026u2026 = -1/12 , I will prove the value of this infinite product : (-2)*(-3)*(-5)*(-7)*(-11)*(-13)*(-17)*u2026u2026u2026u2026. = ?If the mathematician Euler has prove that 1/12 +1/22 +1/32 +1/42 +1/52 +u2026.. =Π 2/6. In this proof , I will generalize this formula for any S , hence S is a complex number Z(S) + Z(-S) = Π 2/6. You will find many other formulas and theorems that justify and prove Riemann hypothesis conjecture.
Category: Number Theory
[3446] viXra:2505.0045 [pdf] submitted on 2025-05-07 19:47:26
Authors: Roberto C. M. Navacchia
Comments: 14 Pages. In Portuguese
The distribution of prime numbers has long intrigued mathematicians, revealing deepstructural patterns within mathematics. This study expands upon Gauss's Circle andGoldbach’s Theorem, focusing on the interaction between prime pairs rather than all naturalnumbers. Through mathematical modeling, persistent gaps within Gauss’s Circle wereidentified, suggesting that it represents the internal structure of the universe, while Goldbach’s prime-counting graph unveils the external structure, formed exclusively by primes. By leveraging Computational Modeling, this work explores these patterns in greater depth, revealing potential underlying rules governing prime distribution. The findings suggest that prime numbers may not only be fundamental to number theory but could also serve as the structural foundation of the universe itself.
Category: Number Theory
[3445] viXra:2505.0044 [pdf] submitted on 2025-05-07 19:45:24
Authors: Samuel Bonaya Buya
Comments: 15 Pages.
This paper presents a unified algebraic, geometric, and analytic framework that redefines the structure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand’s Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings. The paperintroduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros. These results form a bridge betweennumber theory, algebraic topology, mathematical physics, and symbolic computation—offering new tools for understanding prime distributions, factorization, and analytic continuation.
Category: Number Theory
[3444] viXra:2505.0028 [pdf] submitted on 2025-05-05 21:33:15
Authors: Patrick Guiffra
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a novel geometric and analytical framework aimed at addressing the Twin Prime Conjecture, asserting the existence of infinitely many pairs of prime numbers differing by 2, such as (3, 5) and (11, 13).We project prime numbers onto a unit circle, with angles derived from the imaginary parts of the first 100 non-trivial zeros of the Riemann zeta function, defined as θpi = 2π P100n=1 sin(γn ln(pi)) γn mod 2π. By rotating this circle over 100 iterations and generating a binary sequence S(tk) based on a marking interval [0, π 2 , we identify a recurring pattern, "011," with a periodicity of 4 iterations. Numerical simulations across scales up to N = 1024 support this observation,while a formal variance-based contradiction proof argues that this 1 recurrence implies the infinitude of twin primes. A spectral analysis further validates the periodicity, and refined assumptions on the zeta zeros strengthen the theoretical foundation. This work diverges from traditional analytic methods, offering a geometric perspective that emphasizes the need for analytical rigor over numerical scaling.
Category: Number Theory
[3443] viXra:2505.0025 [pdf] submitted on 2025-05-04 14:05:49
Authors: Mustapha Kharmoudi
Comments: 6 Pages.
In this article, we will discuss several properties, both known and novel. Specifically, the genesis of the golden ratio from the Fibonacci sequence. But more importantly, to demonstrate that theFibonacci sequence itself also originates from the very framework of the golden ratio, if I may use that expression. We will reveal new insights into the connection that unites the Fibonaccisequence with the Lucas sequence and Binet’s formula.
Category: Number Theory
[3442] viXra:2505.0012 [pdf] submitted on 2025-05-01 17:38:01
Authors: Younghwan Yun
Comments: 10 Pages.
This paper presents a novel inductive framework for the generation and validation of twin primes, grounded in Bertrand’s Postulate. Unlike traditional methods relying on probabilistic or empirical filtering, this approach provides a recursive structure that not only predicts the location of future twin prime pairs but also supports theoretical generalization to even gaps k = 2,4,6,8,.... Empirical validation up to 109 confirms that no counterexamples violate the proposed inductive inequality conditions. The framework aligns heuristically with the Hardy—Littlewood conjecture and provides evidence supporting both the infinitude and structured distribution of twin primes.
Category: Number Theory
[3441] viXra:2505.0011 [pdf] submitted on 2025-05-01 17:37:28
Authors: Younghwan Yun
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present a constructive and structural proof of the Goldbach Conjecture, which asserts that every even integer greater than two is the sum of two primes. Our approach is based on the concept of factor elimination and prime complement analysis. By categorizing integers into divisors and non-divisors of a given N, and focusing on the structure of non-divisor primes, we demonstrate that the set of complements 2N-a cannot be fully covered by multiples of these primes. Using prime density estimates and structural lemmas, we show that a prime pair (p, q) satisfying p+q=2N must always exist. Numerical examples further validate the framework, providing an intuitive and elementary alternative to heavy analytic methods traditionally used in this domain
Category: Number Theory
[3440] viXra:2505.0010 [pdf] submitted on 2025-05-01 17:36:14
Authors: Younghwan Yun
Comments: 8 Pages.
We present a structural proof of the Collatz conjecture by rigorously analyzing the recursive mapping of odd integers. By introducing a compressed recursive function that directly connects successive odd values, we prove the global injectivity of the sequence and demonstrate that infinite non-repetitive progression is impossible within the constrained domain. We establish that no nontrivial cycles exist through a minimal element argument, and reinforce convergence through nonlinear divergence properties and the Pigeonhole Principle. Consequently, every sequence must inevitably intersect the canonical cycle (1 to 4 to 2 to 1), thus conclusively demonstrating the validity of the Collatz conjecture under the defined structural framework.
Category: Number Theory
[3439] viXra:2505.0009 [pdf] submitted on 2025-05-01 17:34:24
Authors: Younghwan Yun
Comments: 13 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents an intuitive method for proving the Riemann Hypothesis. It begins by deriving the relationship equation at the zeros of the Riemann zeta function from Riemann’s functional equation. This equation follows the Schwarz reflection principle, indicating that the zeros of the zeta function are restricted to the line with a real part of 1/2 in the complex plane. Furthermore, using the Schwarz reflection principle, it concludes that zeros cannot exist outside the critical line. Therefore, the Riemann Hypothesis is true.
Category: Number Theory
[3438] viXra:2505.0001 [pdf] submitted on 2025-05-01 20:48:41
Authors: Samuel Bonaya Buya
Comments: 6 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)
We propose a functional relationship between even and prime numbers that serves as thefoundation for a formal inductive proof framework for the Binary Goldbach Conjecture. This approach is grounded in the principle of prime interval stability, whereby even numbers are represented as functions of pairs of primes constrained within specific intervals. The derived expressions support the conjecture by systematically generating valid Goldbach partitions and establishing inductive continuity in prime pair generation for all even integers greater than two.
Category: Number Theory
[3437] viXra:2504.0189 [pdf] submitted on 2025-04-29 10:21:58
Authors: Julian Beauchamp
Comments: 3 Pages.
In this paper, we observe some nice identities for the products of Fibonacci-like sequences. While these identities can hardly be original discoveries, I have been unable to find them elsewhere.
Category: Number Theory
[3436] viXra:2504.0174 [pdf] submitted on 2025-04-28 20:15:27
Authors: Lautaro Fesembeck
Comments: 10 Pages. Correspondence: lautaro.math@gmail.com
We model the distribution of nontrivial zeros of the Riemann zeta function through a dynamic equilibrium principle. By defining a disturbance field associated with prime distributions and constructing a corresponding global energy functional, we show that any deviation from the critical line Re(s) = 1/2 necessarily increases global energy. Through analysis of local perturbations, global independence, and symmetry properties implied by the functional equation of ζ(s), we demonstrate that only the critical line configuration minimizes total energy. This provides a new and rigorous resolution of the Riemann Hypothesis via energy minimization methods.
Category: Number Theory
[3435] viXra:2504.0173 [pdf] submitted on 2025-04-27 20:05:31
Authors: Christian I. G. Winsor
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a new approach to understanding the Collatz Conjecture. The conjecture asks whether a simple process (repeatedly halving even numbers, and tripling odd numbers then adding one) will always eventually reach the number one, no matter which positive whole number you start with. In this work, I introduce a way to group numbers based on their properties and show that, by following a specific set of steps, every number can be reduced to a smaller group. By combining this method with results that have already been checked by computer for smaller numbers, I provide a logical framework that supports the idea that the Collatz process always ends at one.
Category: Number Theory
[3434] viXra:2504.0164 [pdf] submitted on 2025-04-26 12:50:22
Authors: Immense Raj Subedi
Comments: 5 Pages.
The Collatz Conjecture states that every positive integer eventually reaches 1 through a specific iterative process. This paper presents a novel approach to proving the conjecture by categorizing natural numbers and establishing a key mapping between odd numbers and numbers of a particular form. This structural approach simplifies the problem and provides a comprehensive proof of convergence
Category: Number Theory
[3433] viXra:2504.0160 [pdf] submitted on 2025-04-26 19:20:25
Authors: Abdelmajid Ben Hadj Salem
Comments: 6 Pages.
In this paper, we assume that the explicit abc conjecture of Alan Baker and the conjecture c smaller than R^{1.63} are true, we give a proof of the abc conjecture and we propose the constant K(epsilon)$. Some numerical examples are given.
Category: Number Theory
[3432] viXra:2504.0141 [pdf] submitted on 2025-04-21 20:54:23
Authors: Jochen Kiemes
Comments: 7 Pages.
We present a novel reformulation of the Collatz conjecture by leveraging the binary structure of positive integers, focusing on the sequence of odd terms. Through an analysis of leading and trailing bit-position dynamics, we derive a substantial lower bound of at least 17,026,679,261 steps for any hypothetical non-trivial cycle, offering new insights into its structural constraints.
Category: Number Theory
[3431] viXra:2504.0136 [pdf] submitted on 2025-04-21 17:38:59
Authors: V. Barbera
Comments: 4 Pages.
This paper presents some considerations on the stopping time of the 3n+1 problem. In particular, it presents an algorithm for finding residue classes that have a given stopping time.
Category: Number Theory
[3430] viXra:2504.0134 [pdf] submitted on 2025-04-20 06:54:44
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper reveals a profound mathematical cascade linking three classical number theory phenomena: 1) Bernoulli numbers $B_{n}$ with denominator $6$ ($n equiv 2 pmod{6}$), 2) Special values of Riemann $zeta$-function at even integers, and 3) Enhanced Goldbach partition counts for $x equiv 0 pmod{6}$. We demonstrate their intrinsic connections through von Staudt-Clausen theorem, modular form theory, and statistical verification ($n leq 10^4$, $x leq 10^4$). A $3.2$times$ enhancement ratio in Goldbach partitions emerges as direct consequence of prime number symmetry modulo $6$.
Category: Number Theory
[3429] viXra:2504.0133 [pdf] submitted on 2025-04-20 06:57:07
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a rigorous connection between modular arithmetic constraints and enhanced Goldbach partition counts through dual-channel prime pair combinations. We demonstrate that even numbers $ x equiv 0 pmod{6} $ exhibit $3.2times$ higher partition counts than $ x equiv 2 pmod{6} $ due to symmetric prime distribution modulo 5. The mechanism is visualized through a novel combinatorial diagram (Fig. 1) and supported by statistical analysis ($ x leq 10^4 $).
Category: Number Theory
[3428] viXra:2504.0128 [pdf] submitted on 2025-04-19 08:36:11
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a novel connection between two classical number theory phenomena: 1) Bernoulli numbers $B_n$ with denominator $6$ ($n equiv 2 pmod{6}$) governed by the von Staudt-Clausen theorem, and 2) the enhanced Goldbach partitions for even numbers $x equiv 0 pmod{6}$. We demonstrate their complementary modular symmetry through analytic number theory tools and computational verification. A unified framework is proposed using Rankin-Selberg convolution of modular forms, revealing shared sieve-theoretic mechanisms in prime number distribution.
Category: Number Theory
[3427] viXra:2504.0127 [pdf] submitted on 2025-04-19 08:38:59
Authors: Chenglian Liu, Sonia Chien-I Chen, Ruopengyu Xu
Comments: 2 Pages.
This paper establishes a deep connection between two classical number theory phenomena through modular form-L-function unification: 1) Bernoulli numbers $ B_n $ with denominator 6 ($ n equiv 2 pmod{6} $) governed by von Staudt-Clausen theorem, and 2) enhanced Goldbach partition counts $ G(x) $ for even numbers $ x equiv 0 pmod{6} $. We demonstrate their complementary modular symmetry via:begin{itemize} item Rankin-Selberg convolution of weight-1/weight-2 modular forms item Analytic continuation of associated L-functions item Computational verification ($ n leq 10^4 $, $ x leq 10^4 $)end{itemize}The unified framework reveals that $68.2$% of Bernoulli denominators and $79.4$% of Goldbach enhancements obey modular arithmetic constraints.
Category: Number Theory
[3426] viXra:2504.0125 [pdf] submitted on 2025-04-20 00:26:13
Authors: Piren Mo
Comments: 29 Pages.
Our research has found that all prime numbers share a common new expression, which serves as a necessary but not sufficient condition for a number to be prime. And based on this expression, we have studied the distribution of prime numbers and twin primes, and we are able to predict primes within a certain interval following known primes.
Category: Number Theory
[3425] viXra:2504.0118 [pdf] submitted on 2025-04-17 20:15:36
Authors: Aditya Bagchi
Comments: 21 Pages.
This paper introduces a deterministic framework for validating the conjecture by classifying integers into distinct types based on modulo 16 residues. Positive odd integers are expressed as 16k+m, where m∈{1,3,5,7,9,11,13,15}, representing Types 1 through 8. Positive even integers are expressed as 16k+mu2032, where mu2032∈{0,2,4,6,8,10,12,14} representing EV1 through EV8.The paper considers even numbers as intermediates between two successive odd integers in the Collatz sequence. Under the 3x+1 operation, odd types exhibit distinct divisibility factors (d) that govern their transformations. For instance:AType 1 transforms into Types 1, 3, 5, or 7.B) Type 2 transforms into Types 3 or 7.C) Types 3 and 7 can transform into any odd type.D) Type 4 transforms into Type 2 or 6.E) Type 5 transforms into Type 2, 4, 6 or 8.F) Type 6 transforms into Type 1or 5.G) Type 8 transforms into Type 4 or Type 8 further.Depth First Search (DFS) algorithms identify 911 looping sequences, of which 49 are increasing, and the rest are decreasing. All looping sequences are shown to terminate within finite cycles, and transformations converge universally to 1. The conjecture’s universality is established by proving non-existence of infinite looping and unbound growth. The pigeonhole principle comes into play.
Category: Number Theory
[3424] viXra:2504.0112 [pdf] submitted on 2025-04-18 17:06:40
Authors: Victor Sorokine
Comments: 2 Pages.
The number A + B - C contains unnecessary factor.
Category: Number Theory
[3423] viXra:2504.0110 [pdf] submitted on 2025-04-16 01:52:04
Authors: Jose Acevedo Jimenez
Comments: 3 Pages.
In this article, it is proven that for every n≥3, there exists at least one artificial prime number q such that F_n<q<F_2n , where F_k denotes the k-th number in the Fibonacci sequence. This result is obtained using the Bertrand—Chebyshev theorem and relies on a fundamental property of divisibility within the Fibonacci sequence. Although it does not imply the infinitude of classical primes in the sequence, it does guarantee the existence of infinitely many artificial primes distributed within it.
Category: Number Theory
[3422] viXra:2504.0097 [pdf] submitted on 2025-04-15 18:06:43
Authors: Wing K. Yu
Comments: 11 Pages.
This paper is an improvement on my previous work and proves the Legendre, Oppermann, Brocard, and Andrica conjectures using basic analytical methods.
Category: Number Theory
[3421] viXra:2504.0096 [pdf] submitted on 2025-04-15 21:55:34
Authors: Zhenghao Wu
Comments: 14 Pages.
We provide a general analytic formula to construct all existing starting odd numbers that obey our desired finite arbitrarily long Collatz trajectory, meaning that these starting odd numbers obey our pre-designated maximum factors of 2 at each iteration of the reduced Collatz map. We also provide another general analytic formula for finding the resulting odd numbers after N iterations of the reduced Collatz map. These formulas shed light on the structure of Collatz trajectories and other properties. We can also use this information to find in finite steps all existing Collatz trajectories that become 1 after any finite N iterations.We also will see that the "location" of all of the 1’s in Collatz Conjecture can be found by solving a special case of the discrete log problem.
Category: Number Theory
[3420] viXra:2504.0090 [pdf] submitted on 2025-04-14 19:31:11
Authors: Jayme Mendes
Comments: 11 Pages.
This article presents an algorithm for efficiently generating all prime numbers within the interval [m, n], where m ≥ 3. The algorithm is developed from the demonstration that non-prime numbers in this range can be obtained from certain points in the region between two rectangular hyperbolas and two straight lines. The method does not perform factorization tests and does not require prior knowledge of any prime number, which makes it easier to obtain large primes for n − m = q = constant, when the time and memory complexities become equal to O(log n) and O(1), respectively.
Category: Number Theory
[3419] viXra:2504.0079 [pdf] submitted on 2025-04-12 22:15:36
Authors: Natalia Tanyatia
Comments: 16 Pages. https://github.com/NataliaTanyatia/Optimal-Prime.git (Note by viXra Admin: The article should start with article title, author name and abstract; Please submit article written with AI assistance to ai.viXra.org)
We construct a unified symbolic and geometric framework that links the recursive generation of prime numbers to the problem of closest hypersphere packing in Euclidean space. Beginning with a purely logical definition of primes and building an iterative formula that filters primes based on modular constraints, we establish a symbolic system for exact prime counting and approximation. We then transition from arithmetic to geometry by introducing sphere-packing principles in various dimensions, particularly focusing on both furthest-touching and closest-touching configurations. By analyzing simplex-based Delaunay lattices and maximizing local sphere contact, we show how prime indices emerge naturally as layers in the radial expansion of optimally packed lattices. This construction culminates in a symbolic proof of the Riemann Hypothesis by bounding the prime counting function with a geometric analogy. The result is a cohesive theory in which logical prime filtration, packing density, and analytic continuation of Dirichlet series converge in a single constructively grounded model.
Category: Number Theory
[3418] viXra:2504.0076 [pdf] submitted on 2025-04-11 18:30:49
Authors: Jose Acevedo Jimenez
Comments: 6 Pages.
This paper introduces the concept of artificial primes, defined within a specific subset SSS of positive integers greater than one. A number q∈S is considered an artificial prime if no other element d∈S, with d≠q, divides q. Focusing on the subset of Fibonacci numbers greater than 1, we analyze the behavior of artificial primes in this sequence. Remarkably, the counting function of artificial primes among the first n Fibonacci numbers (with n≥3) matches the classical prime counting function π(n), which enumerates the number of primes less than or equal to n. This correspondence highlights a surprising structural parallel between classical prime distribution and internal divisibility properties within recursive numerical sequences.
Category: Number Theory
[3417] viXra:2504.0072 [pdf] submitted on 2025-04-10 12:43:49
Authors: Rosario D'Amico
Comments: 12 Pages. Keywords: Goldbach's conjecture; Prime number; Probability
This paper aims to provide a set of considerations that allow us to see a possible solution to the problematic issue of Goldbach's "strong" conjecture, which amounts to asserting that any even natural number greater than 2 can be written as the sum of two prime numbers that are not necessarily distinct. Specifically, we will show mathematically that a hypothetical scenario in which no even composite number exists as a sum of two primes is impossible. This will be done by adopting a probabilistic method by far simpler than the arithmetical attempts already present in literature.
Category: Number Theory
[3416] viXra:2504.0065 [pdf] submitted on 2025-04-09 23:37:28
Authors: Samuel Bonaya Buya
Comments: 13 Pages.
This paper explores the relationship between prime gaps and the behavior of the Riemann zeta function. We analyze key logarithmic formulations of the zeta function and their decomposition to real and imaginary parts.A zeta function is formulated that encodes information about Goldbach partitions. An exact prime gap equation is generated. It is shown the prime gaps are determined using the zeroes of the Riemann zeta function. With the exact prime gap formula, an exact formula for counting the number of primes is presented. The mystery of prime numbers is solved.
Category: Number Theory
[3415] viXra:2504.0045 [pdf] submitted on 2025-04-06 22:20:23
Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. Submitted to the journal Bulletin of the London Mathematical Society. Comments welcome.
In this paper, we consider the abc conjecture. Assuming that the conjecture c<rad^{1.63}(abc) is true, we give the proof that the abc conjecture is true.
Category: Number Theory
[3414] viXra:2504.0027 [pdf] submitted on 2025-04-04 22:07:40
Authors: Gabriele Bonamini
Comments: 9 Pages. (Note by viXra Admin: Please cite and list scientific references; Please submit article written with AI assistance to ai.viXra.org)
In this paper, we introduce and develop the concept of "the terminator of a sequence", defined as the term that would be reached by extending the construction of a given infinite succession indefinitely. Unlike the traditional limit, whose value may be abstract or not defined in the target domain, the terminator is an actual element of the sequence.This approach leads to the introduction of new operators which formalize the ideas of "infinity by construction" and "infinitesimal by construction", respectively, which allows to re-design infinite and infinitesimal quantities as concrete mathematical entities that arise directly from the sequence’s construction, rather than as mere abstract mathematical concepts.
Category: Number Theory
[3413] viXra:2504.0021 [pdf] submitted on 2025-04-03 05:37:52
Authors: Song Fei
Comments: 24 Pages.
Abstract This paper introduces the Langlands Watch-Tate (LW-Tate) framework, an extension of the Langlands Watch (LW) framework first proposed in [1] , to prove the Tate Conjecture for all K3 surfaces over mathbb{Q} . We establish that ensuremath{text{rank}text{Pic}(X)=text{ord}_{s=1}L(H^{2}(X),s)} holds universally, covering both finite and infinite automorphism groups, by decomposing H^{2}(X_{overline{mathbb{Q}}},mathbb{Q}_{ell}(1)) into irreducible representations under text{Aut}(X) and associating each with weight 2 automorphic forms on Shimura varieties. Building on LW’s hierarchical structure, LW-Tate’s novel integration of symmetry and modularity resolves a major conjecture in arithmetic geometry. Furthermore, we extend LW-Tate to Calabi-Yau threefolds , explaining text{ord}_{s=2}L(H^{3}(Y),s)=text{rank}text{Pic}(Y) , showcasing its potential to address higher-dimensional Tate Conjectures and cementing its role as a transformative tool in the Langlands Program.
Category: Number Theory
[3412] viXra:2504.0014 [pdf] submitted on 2025-04-02 21:00:51
Authors: Bahbouhi Bouchaib
Comments: 14 Pages.
This article presents two methods A and B for breaking an even number into two primes. Both methods are inspired by the equations 6x ± 1 because all prime numbers and their multiples except 3 are 6x ± 1. Both methods can be very useful for even conversion in sums of two primes or to study the Goldbach's strong conjecture. Both methods can have applications in computer science.
Category: Number Theory
[1875] viXra:2602.0103 [pdf] replaced on 2026-02-23 20:04:59
Authors: Theophilus Agama
Comments: 8 Pages.
Denote the minimal length of a fixed degree d>1 addition chain that leads to n by l^d(n). We introduce the concept of a strong Brauer number of rank d>1 and show that all numbers belonging to this class satisfy the inequality l^d(d^n-1)
Category: Number Theory
[1874] viXra:2602.0024 [pdf] replaced on 2026-02-10 09:39:21
Authors: Sriramadesikan Jagannathan
Comments: 11 Pages. References appended
This paper presents a proof of the Riemann Hypothesis by examining the geometric and arithmetic properties of the Dirichlet eta function. By assuming the existence of zeros off the critical line, and analyzing the resulting alternating series in the complex plane, we establish a logical contradiction. The proof relies on insights into the structure of these series, demonstrating that all non-trivial zeros must possess a real part of exactly 12.
Category: Number Theory
[1873] viXra:2601.0089 [pdf] replaced on 2026-01-27 04:37:39
Authors: Ryan Hackbarth
Comments: 11 Pages. This update includes a more usable primary function, and uses it to forecast the nontrivial zeroes of the Riemann Zeta Function.
Here I present a derivation of an equation whose solution sets are the trivial and nontrivial zeros of the Riemann Zeta Function. I demonstrate how the trivial solutions are directly encoded by integer inputs and how these can be mapped by a symmetry to positive odd integers. I extend this insight to encode the even integers, and map these to the negative odd integers, which provides an explicit connection between particular values of the Riemann Zeta Function which have historical and ongoing research interest. I then extend this symmetry to the nontrivial zeroes, and demonstrate the dependence of the critical line in producing this symmetry. Finally, I note the distribution of the nontrivial zeroes have a correspondence with the distribution of trivial zeroes, and provide a first order approximation of this correspondence.
Category: Number Theory
[1872] viXra:2601.0067 [pdf] replaced on 2026-01-27 02:17:27
Authors: Juan Francisco Petitti
Comments: 40 Pages.
This paper presents a definitive proof of the Riemann Hypothesis by establishing the existence of a self-adjoint operator whose spectrum corresponds precisely to the non-trivial zeros of the Riemann zeta function. Utilizing the framework of quantum mechanics and the strong convergence of a sequence of operators Hn in the supremum norm, we demonstrate that the eigenvalues of the limiting operator are real. We show that the Riemann zeta function is an entire function of order one, satisfying the conditions of the Hilbert-Pólya conjecture. The results confirm that all non-trivial zeros lie on the critical line Re(s)=1/2, thereby resolving the most significant open problem in number theory.
Category: Number Theory
[1871] viXra:2601.0001 [pdf] replaced on 2026-01-25 00:27:26
Authors: Keshava Prasad Halemane
Comments: 16 Pages. 1 Table
This research report presents the Collatz-Hasse-Syracuse-Ulam-Kakutani (CHSUK) Theorem, which asserts the convergence of the Collatz Sequence to the trivial cycle, thus proving the Collatz Conjecture, which has been a long-standing unsolved problem. The proof is based on the bijective isomorphism established between the set of positive integers and a carefully designed system with a hierarchy (arborescence) of binary-exponential-ladders defined on the set of positive odd numbers.
Category: Number Theory
[1870] viXra:2512.0089 [pdf] replaced on 2026-01-28 06:30:48
Authors: Umberto Bartocci, Alessandro Miotto
Comments: 5 Pages.
Given any natural number n, we consider the subset C(n) of all natural numbers which could be written in decimal basis as a string of length n (n-numbers). We are looking for subsets of C(n) which appear interesting in connection with n-prime numbers and twin n-prime numbers too. Those numbers consisting of only the digits 1 and 7 (that we call Sofia numbers) appear to be quite promising in this context, and their study suggests some natural conjecture.
Category: Number Theory
[1869] viXra:2512.0072 [pdf] replaced on 2026-02-09 22:09:31
Authors: Dominique Tremblay
Comments: 71 Pages. Revised version
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p^2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally paired onto the previous one. We also submit to your attention a set of eleven postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1868] viXra:2512.0072 [pdf] replaced on 2026-01-20 01:04:46
Authors: Dominique Tremblay
Comments: 60 Pages. Revised version
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of eleven postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1867] viXra:2512.0072 [pdf] replaced on 2025-12-21 18:54:43
Authors: Dominique Tremblay
Comments: 58 Pages. Revised version
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of ten postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1866] viXra:2512.0064 [pdf] replaced on 2026-02-09 22:09:57
Authors: Dominique Tremblay
Comments: 73 Pages. French version / Revised version
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p^2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally paired onto the previous one. We also submit to your attention a set of eleven postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1865] viXra:2512.0064 [pdf] replaced on 2026-01-20 01:02:37
Authors: Dominique Tremblay
Comments: 66 Pages. French version
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of eleven postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1864] viXra:2512.0064 [pdf] replaced on 2025-12-21 20:57:08
Authors: Dominique Tremblay
Comments: 62 Pages. In French
In this article, we intend to present two procedures of elementary simplicity, one for predicting the complete sequence of prime numbers, the second being more specifically dedicated to the identification of pairs of twin primes, with the primary motivation of creating the most concise method of achieving this end. To detect prime numbers in the continuum of natural numbers, our study is inspired by an algorithm well known to mathematicians working in the field, namely the formula p2 = 24n + 1, to which we grafted two additional sieving steps aimed at discriminating false positives. The procedure for predicting the specific sequence of twin primes can be conceived independently or operationally grafted onto the previous one. We also submit to your attention a set of ten postulates concerning twin primes, as well as a sample of eleven additional hypothesis of the order of conjecture.
Category: Number Theory
[1863] viXra:2511.0133 [pdf] replaced on 2025-12-09 01:08:30
Authors: Ryan Hackbarth
Comments: 5 Pages. (Note by viXra Admin: An abstract is required in the article)
In this paper, I utilize Euler's derivation of the Product Formula the Zeta function to produce a similar Product Formula for the Dirichlet Eta Function. I then examine how this formula relates to the critical line and the zeros of the Riemann Zeta Function.
Category: Number Theory
[1862] viXra:2511.0093 [pdf] replaced on 2025-12-31 02:21:34
Authors: Immense Raj Subedi
Comments: 13 Pages.
This paper presents a novel approach to the Collatz conjecture by focusing on the subsetof natural numbers expressed in the form 12n − 4. By analyzing the algebraic mappingsand trajectories of these numbers under the Collatz function, we demonstrate that theirsequences remain within this form and exhibit a strictly decreasing behavior. We establishthat the transformations lead to a pipeline of values that map back to smaller terms of thesame form. Crucially, we provide a **rigorous algebraic proof of net descent** for all fourcongruence classes modulo 4, including the previously challenging cases of initial growth.This proof ensures the absence of non-trivial cycles and guarantees convergence to 1. Sinceevery natural number eventually reaches an odd number, and the odd numbers correspondto this subset via our mapping, the results imply and **establish the convergence of allnatural numbers to 1, providing a complete proof of the conjecture.**
Category: Number Theory
[1861] viXra:2511.0024 [pdf] replaced on 2025-11-23 17:36:46
Authors: Satya Das
Comments: 22 Pages. The content is changed as part of an improvement
We establish a structural correspondence between the Collatz map and the signedJacobsthal numbers, providing an arithmetic reformulation of the 3x + 1 problem. Byrepresenting Collatz iterations through powers of signed Jacobsthal numbers, we derivenecessary and sucient conditions for the existence of cycles and for the validity ofthe coecient stopping time conjecture. This formulation translates the combinatorialdynamics of the Collatz map into explicit number-theoretic identities, revealing anunderlying algebraic framework that connects iteration, recurrence, and integrality.The results suggest a pathway toward analyzing the conj
Category: Number Theory
[1860] viXra:2510.0135 [pdf] replaced on 2025-12-12 01:08:37
Authors: Zihang Chen
Comments: 10 Pages.
This paper will start with the derivation of the Euler-Maclaurin formula with singularities, compensate for the series divergence problem of its fitting the zeta function by adding a compensation factor ε, perform analytic continuation on the expanded series, analyze the distribution of its trivial zeros through the laws of Bernoulli numbers, and then construct functions and combine the properties of the gamma function to solve the distribution law of non-trivial zeros.
Category: Number Theory
[1859] viXra:2510.0135 [pdf] replaced on 2025-11-06 02:47:36
Authors: Chen Zihang
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper will expand the Riemann zeta function via the Euler-Maclaurin formula, analyze it together with the Riemann zeta function after analytic continuation, and derive a new formula describing the behavior of the zeta function. By analyzing this new formula related to Bernoulli numbers, the distribution law of its trivial zeros will be identified. Furthermore, the infinite product form of the gamma function and the analytic continuation definition of the Riemann zeta function will be used to find the distribution law of non-trivial zeros, ultimately proving the Riemann Hypothesis that Re(s) = 1/2.
Category: Number Theory
[1858] viXra:2510.0052 [pdf] replaced on 2025-11-15 03:41:26
Authors: Hashem Sazegar
Comments: 7 Pages.
Oppermann’s conjecture states that for every positive integer n, there exists at least one prime number between n 2 and n 2 + n. Priorto this, Legendre had conjectured that there is always at least one prime number between n2 and (n + 1)2 . In this paper, we not onlyclaim to prove Oppermann’s conjecture but also propose a smaller interval, asserting that there exists at least one prime between n 2 andn 2 + n/2.
Category: Number Theory
[1857] viXra:2509.0005 [pdf] replaced on 2025-09-10 16:08:44
Authors: Jabari Zakiya
Comments: 14 Pages. New content added before publication.
Various bounds on p, such as Bertrand’s Postulate and Legendre’s Conjecture, propose regions around n that have at at least one prime within them. Using Prime Generator Theory, I show more precise symmetric bounds on p, such that for n a prime exists symmetrically within a distance of n^(1/2) below and above it. That is to say, a prime exists for: n — n^(1/2) < p < n and n < p < n + n^(1/2).
Category: Number Theory
[1856] viXra:2508.0170 [pdf] replaced on 2025-10-17 23:36:16
Authors: Abdelhay Benmoussa
Comments: 7 Pages.
Let $Vf(x) = int_0^x f(t),dt$ denote the Volterra operator. We derive an explicit expansion for the iterated operator $(xV)^n$ in terms of powers of $V$:$(xV)^n = sum_{k=0}^{n-1} (-1)^k a(n-1,k), x^{,n-k} V^{,n+k},$where $a(n,k)$ are the Bessel coefficients (OEIS A001498). This identity may be viewed as an integral analogue of the classical Grunert's operational formula$(xD)^n = sum_{k=0}^n S(n,k), x^k D^k,$where $S(n,k)$ are the Stirling numbers of the second kind. We also obtain a closed integral representation for $(xV)^n$ and give two applications illustrating the operator identity.
Category: Number Theory
[1855] viXra:2508.0170 [pdf] replaced on 2025-09-26 22:58:33
Authors: Abdelhay Benmoussa
Comments: 10 Pages.
We show that the operator (xV)^n can be expressed as a sum of powers of the Volterra operator $V$, with coefficients given by the Bessel coefficients A001498.
Category: Number Theory
[1854] viXra:2508.0170 [pdf] replaced on 2025-09-01 23:18:31
Authors: Abdelhay Benmoussa
Comments: 13 Pages. Some refinements
We study the integral analog of the operator (left(x frac{d}{dx}ight)^n), obtained by replacing differentiation with integration. We prove that the resulting operator admits an expansion in powers of the integration operator with coefficients given by the Bessel numbers of the second kind ({B(n,k)}) (OEIS seqnum{A122848}), leading to new explicit formulas and revealing a fundamental role of Bessel numbers in the structure of certain integral operators. The author has contributed these findings to the corresponding entries in the OEIS.
Category: Number Theory
[1853] viXra:2508.0165 [pdf] replaced on 2026-01-19 00:30:18
Authors: Teo Banica
Comments: 400 Pages.
This is an introduction to numbers, fractions, percentages and arithmetic. We first discuss what can be done with integers and their quotients, namely basic arithmetic, all sorts of counting results, and with a look into abstract algebra and quadratic residues too. We then upgrade our knowledge by introducing the real numbers, and exploring what can be done with them, in relation with number theory questions. Then we further upgrade our methods, by introducing and using the complex numbers. Finally, we provide an introduction to the zeta function, and the Riemann hypothesis.
Category: Number Theory
[1852] viXra:2508.0067 [pdf] replaced on 2025-08-22 02:31:43
Authors: John Yuk Ching Ting
Comments: 46 Pages. Proofs for Generalized Riemann hypothesis, Birch and Swinnerton-Dyer conjecture, and Polignac's and Twin prime conjectures
Whereby all infinitely-many prime numbers are classified as [well-defined] Incompletely Predictable entities, so must all infinitely-many nontrivial zeros be classified as such. We outline the interesting observations and conjectures about distribution of nontrivial zeros in L-functions; and [optional] use of Sign normalization when computing Hardy Z-function, including its relationship to the Analytic rank and Symmetry type of L-functions. When Sign normalization is applied to L-functions, we posit its dependency on even-versus-odd Analytic ranks, degree of L-function, and particular gamma factor present in functional equations for Genus 1 elliptic curves and higher Genus curves. By invoking inclusion-exclusion principle, our mathematical arguments are postulated to satisfy Riemann hypothesis, and Birch and Swinnerton-Dyer conjecture in their Generalized formats. We explicitly mention underlying proven / unproven hypotheses or conjectures. We provide Algebraic-Transcendental proof (Proof by induction) as supplementary material for open problem in Number theory of Riemann hypothesis whereby it is proposed all nontrivial zeros of Riemann zeta function are located on its Critical line.
Category: Number Theory
[1851] viXra:2508.0057 [pdf] replaced on 2025-08-22 02:34:27
Authors: John Yuk Ching Ting
Comments: 36 Pages. Proofs for Generalized Riemann hypothesis, Birch and Swinnerton-Dyer conjecture, and Polignac's and Twin prime conjectures
Utilizing Input-White Box-Output (I-WB-O) Modeling, we outline novel applications of Mathematics for Incompletely Predictable Problems (MIPP) to unique sets and subsets from prime and composite numbers, nontrivial zeros of Riemann zeta function, and selected number sequences from The On-Line Encyclopedia of Integer Sequences (OEIS). We show that MIPP is valid for mathematical functions, equations or algorithms containing an equality relationship between two expressions. When applied to OEIS number sequence A228186, MIPP is also valid for an inequality. Inclusion-exclusion (I-E) principle from combinatorics removes contributions from over-counted elements in sets and subsets. Invoking I-E principle, arising consequences from MIPP formulations containing I-WB-O Models provide necessary mathematical arguments for rigorously solving open problems Riemann hypothesis, Polignac's and Twin prime conjectures.
Category: Number Theory
[1850] viXra:2508.0018 [pdf] replaced on 2025-08-29 12:59:28
Authors: V. Barbera
Comments: 13 Pages.
This paper presents an analysis of the stopping time of the 3n+1 problem based on the residue class of n.
Category: Number Theory
[1849] viXra:2508.0018 [pdf] replaced on 2025-08-08 14:10:49
Authors: V. Barbera
Comments: 18 Pages.
This paper presents an analysis of the stopping time of the 3n+1 problem based on the residue class of n.
Category: Number Theory
[1848] viXra:2507.0190 [pdf] replaced on 2025-11-16 19:33:56
Authors: Marcin Barylski
Comments: 7 Pages. Updating results and the greatest prime island found so far (of size 16).
There are several interesting ways to depict distribution of primes like Ulam Spiral, Klauber Triangle or the Sacks Number Spiral. In all cases, Prime Number Theorem describes the asymptotic distribution of such numbers among the positive integers. This work is devoted to illustration of primes of form p x q +- C in a way that allows to search for clusters (so called islands of primes). The direct goal of this experimental work is to locate islands with the largest surface area and potentially discover some further patterns in distribution of primes.
Category: Number Theory
[1847] viXra:2507.0190 [pdf] replaced on 2025-10-26 20:15:45
Authors: Marcin Barylski
Comments: 6 Pages. Adding more experiments/results.
There are several interesting ways to depict distribution of primes like Ulam Spiral, Klauber Triangle or the Sacks Number Spiral. In all cases, Prime Number Theorem describes the asymptotic distribution of such numbers among the positive integers. This work is devoted to illustration of primes of form p x q +- C in a way that allows to search for clusters (so called islands of primes). The direct goal of this experimental work is to locate islands with the largest surface area and potentially discover some further patterns in distribution of primes.
Category: Number Theory
[1846] viXra:2507.0186 [pdf] replaced on 2025-12-23 09:28:55
Authors: Bo Zhang
Comments: 10 Pages.
The Riemann hypothesis is proved true by finding two linear combinations of the real and imaginary parts of the Riemann $xi$-function.
Category: Number Theory
[1845] viXra:2507.0186 [pdf] replaced on 2025-10-02 01:15:06
Authors: Bo Zhang
Comments: 10 Pages.
The Riemann hypothesis is proved true by finding two linear combinations of the real and imaginary parts of the Riemann x-function
Category: Number Theory
[1844] viXra:2507.0186 [pdf] replaced on 2025-07-27 03:59:37
Authors: Bo Zhang
Comments: 4 Pages.
The Riemann hypothesis is proved true through a linear combination of the real and imaginary parts of the Riemann $xi$-function.
Category: Number Theory
[1843] viXra:2507.0143 [pdf] replaced on 2025-12-15 02:13:35
Authors: Fawang Su
Comments: 15 Pages.
By setting the 3x + 1 iterative exponential orbit to construct and solve the initial number, and allowing the running orbit of this initial number to iterate under the control of the exponential orbit, this construction method proves that the step length of the 3x + 1 iterative orbit for specific types of numbers can be arbitrary or even infinitely long.
Category: Number Theory
[1842] viXra:2507.0143 [pdf] replaced on 2025-08-09 03:48:08
Authors: Fawang Su
Comments: 15 Pages.
By setting the even - exponent orbit of the 3x + 1 iteration to construct and solve the initial number, and let the odd - orbit of this initial number iterate under the control of the even - exponent orbit. Through this construction method, it is proved that the 3x + 1 iteration step length of a specific type of number can be arbitrary or even infinite.
Category: Number Theory
[1841] viXra:2507.0135 [pdf] replaced on 2025-07-25 21:35:53
Authors: Kuan Peng
Comments: 22 Pages.
Pythagorean triples are generated with Euclid’s formula. But how this formula was derived by or before Euclid is a mystery. We have derived Euclid’s formula directly from Pythagorean equation and classified all Pythagorean triples in a 3D table. The equation X^2+Y^j=Z^2 is proven to have infinitely many integer solutions. By comparing Pythagorean equation with Fermat’s equation for n=3 we were able to explain why Fermat’s equation with n=2 has integer solutions while with n 3 it has not. We propose an algebraic method to work Fermat’s last theorem.
Category: Number Theory
[1840] viXra:2507.0134 [pdf] replaced on 2025-12-31 02:45:26
Authors: Izzie Boxen
Comments: 64 Pages.
The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems being proved. Included is a new and independent proof for Bertrand’s postulate, two proofs for Lim((p_[n+1]-p_n)/p_n)=0, and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann.
Category: Number Theory
[1839] viXra:2507.0134 [pdf] replaced on 2025-10-06 03:23:01
Authors: Izzie Boxen
Comments: 63 Pages.
The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems being proved. Included is a new and independent proof for Bertrand’s postulate. Based on a conjecture on the distribution of prime candidates, proofs for lim(p_[n+1]-p_n)/p_n = 0, and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann are provided.
Category: Number Theory
[1838] viXra:2507.0134 [pdf] replaced on 2025-07-31 20:32:09
Authors: Izzie Boxen
Comments: 63 Pages.
The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems and some extant conjectures being proved. Included are a new and independent proof for Bertrand’s postulate, proofs for lim((p_[n+1]-p_n)/p_n)=0, and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann.
Category: Number Theory
[1837] viXra:2507.0049 [pdf] replaced on 2025-07-17 20:32:38
Authors: Denis Micheal Odwar
Comments: 8 Pages.
For m ∈ Z, let N = 2m ≥ 8 and GN be a set of goldbach primes, p, of N defined as GN = {p : p ≤ N 2 andp ∤ N}. By denoting the cardinality of GN by |GN| or g(N), we show that ∀N, |GN| > 0, and the set of all these cardinalities, {|GN|}, is equal to the set of natural numbers i.e {|GN|} = N = {1,2,3,4,5,6,···}. We finally prove the famous binary |GN| Goldbach conjecture by showing that for all values of |GN|, i=1 µ(bi)Λ(bi) < 0, whenever bi = N −pi with pi ∈ GN,i ∈ N and 1 ≤ i ≤ |GN|. In particular we show that every N is a sum of two distinct primes.
Category: Number Theory
[1836] viXra:2507.0049 [pdf] replaced on 2025-07-09 15:01:25
Authors: Denis Micheal Odwar
Comments: 9 Pages.
For m ∈ Z, let N = 2m ≥ 8 and GN be a set of goldbach primes, p, of N defined as GN = {p : p ≤ N 2 andp ∤ N}. By denoting the cardinality of GN by | GN | or g(N), we show that ∀N, | GN |> 0, and the set of these cardinalities, {| GN |}, is equal to the set of natural numbers i.e {| GN |} = N. We finally prove the famous Goldbach binary |GN| conjecture by showing that i=1 µ(bi)Λ(bi) < 0, whenever bi = N −pi with pi ∈ GN,i ∈ N and 1 ≤i≤| GN | . In particular we show that every N is a sum of two distinct primes.
Category: Number Theory
[1835] viXra:2507.0020 [pdf] replaced on 2025-11-22 00:23:52
Authors: Chloe Williams
Comments: 9 Pages.
In this paper, I will prove all positive odd integers link incrementally through connected orbits to a value 2^c where c ∈ N. These orbits connect because of a function I defined based on the connection between all odd and 8mod12 positive integers. Applying this function infinitely many times to all positive odd integers always leads to a connection to power of 2 orbits. With these results, I’m able to prove the conjecture.
Category: Number Theory
[1834] viXra:2506.0121 [pdf] replaced on 2026-02-26 17:06:52
Authors: Jose Risomar Sousa
Comments: 12 Pages. A typo in the very final formula has been fixed.
A generalization of the Riemann functional equation with a broader validity domain than the one available in the literature is introduced. The insight that led to this new relation came from a new formula for the zeta function created herein that implies the Riemann functional equation. Further developments that stem from new formulae introduced previously are also discussed.
Category: Number Theory
[1833] viXra:2506.0121 [pdf] replaced on 2025-07-16 01:45:35
Authors: Jose Risomar Sousa
Comments: 12 Pages.
A generalization of the Riemann functional equation with a broader validity domainthan the one available in the literature is introduced. The insight that led to this newrelation came from a new formula for the zeta function created herein that implies theRiemann functional equation. Further developments that stem from new formulae introducedpreviously are also discussed.
Category: Number Theory
[1832] viXra:2506.0121 [pdf] replaced on 2025-06-27 01:10:19
Authors: Jose Risomar Sousa
Comments: 11 Pages.
A generalization of the Riemann functional equation with a broader validity domain than the one available in the literature is introduced. The insight that led to this new relation came from a new formula for the zeta function created herein that implies the Riemann functional equation. Further developments that stem from new formulae introduced previously are also discussed.
Category: Number Theory
[1831] viXra:2506.0079 [pdf] replaced on 2025-06-23 00:42:25
Authors: Walter A. Kehowski
Comments: 8 Pages. The abstract was rewritten, a proof of the Euclid-Euler Theorem was included, and some typos corrected. Comments welcome.
A sublime number is a number such that both its number of divisors and sum of divisors are both perfect numbers. The number 12 is the first sublime number. This paper gives Kevin Brown's construction of even sublime numbers a modern mathematical development.
Category: Number Theory
[1830] viXra:2506.0052 [pdf] replaced on 2025-12-14 05:44:37
Authors: Hajime Mashima
Comments: 82 Pages.
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[1829] viXra:2506.0052 [pdf] replaced on 2025-11-23 01:22:55
Authors: Hajime Mashima
Comments: 82 Pages.
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[1828] viXra:2506.0052 [pdf] replaced on 2025-08-31 11:26:50
Authors: Hajime Mashima
Comments: 81 Pages.
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[1827] viXra:2506.0052 [pdf] replaced on 2025-06-29 02:09:08
Authors: Hajime Mashima
Comments: 81 Pages. In Japanese
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[1826] viXra:2506.0019 [pdf] replaced on 2025-06-15 01:54:23
Authors: Hans Hermann Otto
Comments: 9 Pages.
We show the importance of Lucas number 123, symbolized by others for divine order or trinity, as enigmatic number for life, physics and the cosmos. This number is related to fundamental constants of nature and to powers of the golden mean that governs phase transitions from particle to cosmic scale. It is also related to the Higgs boson and to the Great Pyramid. Our number theoretical approach may help to understand unsolved problems in physics and supports applications.
Category: Number Theory
[1825] viXra:2506.0001 [pdf] replaced on 2025-07-23 20:25:30
Authors: Mar Detic
Comments: 3 Pages.
We introduce a primality testing framework based on examining the greatest common divisors (GCDs) of a candidate integer p with terms froma quadratic sequence defined by Sq = q2 + 2q for odd integers q. This approach generalizes and extends classical primality criteria by leveraging properties of quadratic forms. We formalize this method as a conjecture,analyze its computational complexity, discuss potential error cases such as pseudoprimes, and present empirical validations demonstrating its effectiveness.
Category: Number Theory
[1824] viXra:2506.0001 [pdf] replaced on 2025-06-03 23:52:15
Authors: Mar Detic
Comments: 3 Pages.
This document introduces and investigates a criterion for defining primality within the sequence of odd integers Ak = 2k + 1. The criterion is based on the greatest common divisor (GCD) between a term Ak and the preceding partial sums Sj = j2 + 2j of the same sequence. We formally define the sequence,its partial sums, and the proposed primality criterion. Computational observations suggest that all standard prime numbers within the sequence are classified as "prime" by this definition. Furthermore, computational observations suggest that all composite numbers in the sequence are classified as "notprime," including Carmichael numbers, which are known for their pseudoprime properties. This leads to a conjecture that the proposed criterion is equivalent to the standard definition of primality for terms in this specific arithmetic progression. Examples are provided to illustrate the application of the criterion for both prime and composite numbers, including a Carmichael number. A further conjecture is made regarding a potential computational bound for verification, leading to a discussion of its complexity.
Category: Number Theory
[1823] viXra:2505.0206 [pdf] replaced on 2025-06-08 21:28:17
Authors: Younghwan Yun
Comments: 7 Pages.
This paper presents a structural proof that any number satisfying the internal additive and multiplicative symmetries of a perfect number must be even. By decomposing the proper divisors of a perfect number into two ordered subsets, we derive a recursive system of proportional identities. We show that this system admits integer solutions only when all proportional coefficients equal one, thereby forcing the smallest divisor to be two. This structural condition excludes the possibility of odd perfect numbers under the proposed model. Our approach not only supports the longstanding conjecture that all perfect numbers are even but also provides a generalized framework thatmay extend to the analysis of semiperfect and abundant numbers.
Category: Number Theory
[1822] viXra:2505.0204 [pdf] replaced on 2025-09-24 16:09:42
Authors: Abdelmajid Ben Hadj Salem
Comments: 3 Pages.
In this paper, we assume that the explicit abc conjecture of Alan Baker (2004) is true, and prove that c smaller than rad^{2}(abc) is true; it is one of the keys to resolve the mystery of the abc conjecture.
Category: Number Theory
[1821] viXra:2505.0186 [pdf] replaced on 2025-06-05 20:05:29
Authors: Junho Eom
Comments: 14 Pages. 2 figures
This paper identified the characteristics of prime numbers within a limited boundary, defined primes between quadratic intervals, and generalized Legendre’s conjecture. Regarding the boundary, every integer less than m was defined as the 1st boundary and it expanded to the mth boundary within m^2. Thus, each boundary contained m elements. Except for 1, every integer produced a sine wave from the 1st boundary; as a result, only prime waves affected the remaining boundaries from the 2nd to mth by generating the composites and new primes (Series I). Therefore, the number of new primes in each boundary could not exceed PI(1st boundary) or PI(m), where PI(x) was the number of primes less than or equal to x, and it enabled the estimation of the total number of primes within m^2 (Series II). Based on Series I and II, the quadratic intervals between PI(m^2) and PI((m + 1)^2) were identical to the sum of the last two boundaries, expressed as 2·βm+1·PI(m), where βm+1 was the ratio of PI((m + 1)^2) to PI(m + 1)·(m + 1) (Series III). This led to the conclusion that Legendre’s conjecture satisfied while 0.8986·PI(P) < 2·βP·PI(P) < PI(P) (prime P > 113), or 2·βm+1·PI(m) ≤ 2·PI(m) (integer m ≥ 2).
Category: Number Theory
[1820] viXra:2505.0179 [pdf] replaced on 2025-09-20 22:59:42
Authors: Khalid ibraheem Al-Ibraheem
Comments: 14 Pages.
Addressing the notorious difficulty of this problem, Richard Guy (1) once advised: "don’t try to solve these problems!"
In this paper we have performed the function to specific sets of odds C, D, and E:
C = ∑i=0b-1 4i + 2⋅4b⋅n | b ≥ 1, n ≥ 0, where f(C) = 1 + 6n
D = 3 + 4⋅n | n ≥ 0, where f(D) = 5 + 6n
E = 3 + 10⋅(∑i=0b-1 4i) + 4(b+1)⋅n | b ≥ 1, n ≥ 0, where f(E) = 5 + 6n
We subsequently prove that all integers return to 1 under the iteration of the Collatz function by analyzing the behavior of set V = 5 + 12n.
Category: Number Theory
[1819] viXra:2505.0150 [pdf] replaced on 2025-05-29 08:46:39
Authors: Bahbouhi Bouchaib
Comments: 11 Pages. The paper shows new data about Goldbach's strong conjecture
In this article I apply classical statistical laws to analyze prime numbers assimilated to populations. The statistical analysis focuses on prime numbers in the intervals [0 - S/2] and [S/2 — S] with S an even > 4. The results show that the even number S > 4 is enclosed by two populations of prime numbers P in the interval [0 - S/2] and Q in [S/2 - S] which have approximately the same standard deviation relative to their means. Two other subpopulations P' included in P and Q' included in Q which satisfy the Goldbach's strong conjecture (P' + Q' = S) also have the same standard deviation and superimpose or overlap. This result shows that an even number is enclosed by two populations P' and Q' of prime numbers which are symmetric with respect to S/2 and therefore S = P' + Q'. This result also shows that any natural number N > 4 is enclosed by at least two equidistant and symmetric prime numbers. Therefore for every N > 4 there exists a number t < N such that N — t = P' and N + t = Q' are primes and so 2N = P' + Q'.
Category: Number Theory
[1818] viXra:2505.0110 [pdf] replaced on 2025-05-22 16:12:15
Authors: Bube Ibekwe
Comments: 6 Pages.
This paper investigates the variance of transformed twin primes, where each prime pair (p, p+2) is mapped to k = (p+1)/6. Initial analysis suggested unexpected growth patterns in the normalized variance, seemingly contradicting theoretical expectations. Through large-scale computation of twin primes up to one billion, we demonstrate that these apparent anomalies arise from normalization artifacts. The true variance follows a quadratic growth pattern, with our empirical results closely matching predicted scaling behavior. We resolve the paradox by showing how the sparse distribution of twin primes distorts normalized statistical measures. Our findings highlight critical pitfalls in analyzing prime distributions and provide new insights into the statistical behavior of twin primes.
Category: Number Theory
[1817] viXra:2505.0044 [pdf] replaced on 2025-05-16 22:31:29
Authors: Samuel Bonaya Buya
Comments: 37 Pages.
This paper presents a unified algebraic, geometric, and analytic framework that redefines thestructure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand’s Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings.The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros.In addition to these foundations, the paper offers resolved formulations of three major number theoryconjectures: (1) a short proof of Beal’s Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica’s Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework,unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation—offering new tools for understanding prime distributions, factorization, and analytic continuation.
Category: Number Theory
[1816] viXra:2505.0012 [pdf] replaced on 2025-06-19 21:10:21
Authors: Younghwan Yun
Comments: 13 Pages.
The Twin Prime Conjecture asserts the infinitude of prime pairs (p,p + 2). While recent breakthroughs by Zhang, Maynard, and Tao have demonstrated the infinite occurrence of bounded prime gaps, they fall short of resolving the specific case of gap 2. This paper proposes a structural framework that directly addresses the twin prime case through a modular, sieve-based approach. We demonstrate that twin prime candidates of the form (6k − 1,6k + 1) persist indefinitely under periodic sieving, supported by the inclusion-exclusion principle and recursive inductive logic derived from Bertrand’s Postulate. Unlike probabilistic or density-based methods, our approach emphasizes logical irreducibility and structural necessity. We formalize this persistence through a series of lemmas and prove that no finite sieve can entirely eliminate such candidates. Computational tests up to 109 confirm the validity of the inductive conditions not only for the canonical gap k = 2 but also for larger even gaps k = 4,6,8,10, supporting a generalization aligned with Polignac’s Conjecture. These findings suggest that twin primes, and more broadly even-gapped prime pairs, are an inevitable outcome of arithmetic structure rather than statistical anomaly.
Category: Number Theory
[1815] viXra:2505.0011 [pdf] replaced on 2025-07-03 00:12:43
Authors: Younghwan Yun
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We propose a structural and combinatorial proof of the Goldbach Conjecture, asserting that every even integer greater than two can be written as the sum of two primes. The approach introduces a dual-layer framework. First, we quantify the number of composite pairs that could obstruct the formation of valid Goldbach partitions by systematically classifying and eliminating non-prime candidates arising from non-divisor prime multiplicities. This quantitative asymmetry reveals that the available prime candidates on the complementary side of the partition always outnumber the obstructive composites. Second, we introduce a structural decomposition that constructs prime complements from non-divisor primes and rigorously shows that these complements cannot be fully covered by composite multiples of the base primes. As a result, at least one uncovered and irreducible complement must be a prime, guaranteeing the existence of a valid prime pair. This hybrid method bridges enumerative and structural perspectives, providing an elementary yet rigorous proof route that avoids traditional analytic machinery and reveals inherent prime-generating asymmetries within the even number structure.
Category: Number Theory
[1814] viXra:2505.0011 [pdf] replaced on 2025-05-21 20:08:34
Authors: Younghwan Yun
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present a constructive and structural proof of the Goldbach Conjecture, which asserts that every even integer greater than two is the sum of two primes. Our approach is based on the concept of factor elimination and prime complement analysis. By categorizing integers into divisors and non-divisors of a given N, and focusing on the structure of non-divisor primes, we demonstrate that the set of complements 2N-a cannot be fully covered by multiples of these primes. Using prime density estimates and structural lemmas, we show that a prime pair (p, q) satisfying p+q=2N must always exist. Numerical examples further validate the framework, providing an intuitive and elementary alternative to heavy analytic methods traditionally used in this domain.
Category: Number Theory
[1813] viXra:2505.0010 [pdf] replaced on 2025-06-04 22:02:18
Authors: Younghwan Yun
Comments: 11 Pages.
We present a structural proof of the Collatz conjecture by rigorously analyz-ing the recursive mapping of odd integers. By introducing a compressed recursive functionthat directly connects successive odd values, we prove that the trajectory of the sequenceis globally non-repeating within the constrained domain. We establish that no nontrivialcycles exist through a minimal element argument, and reinforce convergence through nonlin-ear divergence properties and the Pigeonhole Principle. Consequently, every sequence mustinevitably intersect the canonical cycle (1 → 4 → 2 → 1), thus conclusively demonstratingthe validity of the Collatz conjecture under the defined structural framework.
Category: Number Theory
[1812] viXra:2505.0010 [pdf] replaced on 2025-05-21 20:10:28
Authors: Younghwan Yun
Comments: 12 Pages.
We present a structural proof of the Collatz conjecture by rigorously analyzing the recursive mapping of odd integers. By introducing a compressed recursive function that directly connects successive odd values, we prove the global injectivity of the sequence and demonstrate that infinite non-repetitive progression is impossible within the constrained domain. We establish that no nontrivial cycles exist through a minimal element argument, and reinforce convergence through nonlinear divergence properties and the Pigeonhole Principle. Consequently, every sequence must inevitably intersect the canonical cycle (1 → 4 →2→1), thus conclusively demonstrating the validity of the Collatz conjecture under the defined structural framework.
Category: Number Theory
[1811] viXra:2504.0160 [pdf] replaced on 2025-11-16 00:10:23
Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages. The proof is corrected.
In this paper, we assume that the explicit abc conjecture of Alan Baker and the conjecture c [smaller than] R^{1.63} are true, we give a proof of the abc conjecture and we propose the constant K(epsilon). Some numerical examples are provided.
Category: Number Theory
[1810] viXra:2504.0136 [pdf] replaced on 2025-05-01 17:22:42
Authors: V. Barbera
Comments: 6 Pages.
This paper presents some considerations on the stopping time of the 3n+1 problem. In particular, it presents an algorithm for finding residue classes that have a given stopping time.
Category: Number Theory
[1809] viXra:2504.0097 [pdf] replaced on 2025-10-27 23:58:58
Authors: Wing K. Yu
Comments: 13 Pages.
This paper improves my previous proof of the Legendre conjecture by reducing some redundant statements, improving some corollaries, and simplifying two data tables.
Category: Number Theory
[1808] viXra:2504.0096 [pdf] replaced on 2025-04-18 01:19:02
Authors: Zhenghao Wu
Comments: 21 Pages.
We provide a general analytic formula to construct all existing starting odd numbers that obey our desired finite arbitrarily long Collatz trajectory, meaningthat these starting odd numbers obey our pre-designated maximum factors of 2 at each iteration of the reduced Collatz map. We also provide another generalanalytic formula for finding the resulting odd numbers after N iterations of the reduced Collatz map. These formulas shed light on the structure of Collatztrajectories and other properties. We can also use this information to find in finite steps all existing Collatz trajectories that become 1 after any finite N iterations.We also will see that the "location" of all of the 1’s in Collatz Conjecture can be found by solving a special case of the discrete log problem.
Category: Number Theory
[1807] viXra:2504.0090 [pdf] replaced on 2025-04-22 12:11:40
Authors: Jayme Mendes
Comments: 10 Pages.
This article presents an algorithm for efficiently generating all prime numbers within the interval $[m,n]$, where $mgeq 3$. The algorithm is developed from the demonstration that non-prime numbers in this range can be obtained from certain points in the region between two rectangular hyperbolas and two straight lines. The method does not perform factorization tests and does not require prior knowledge of any prime number, which makes it easier to obtain large primes for $n-m=q=constant$, when the time and memory complexities become equal to ${cal O}(log n)$ and ${cal O}(1)$, respectively.
Category: Number Theory
[1806] viXra:2504.0072 [pdf] replaced on 2025-08-20 20:27:57
Authors: Rosario D'Amico
Comments: 12 Pages. https://dx.doi.org/10.23755/rm.v54i0.1662 keywords: Goldbach' Conjecture, Prime numbers, Probability
This paper aims to provide a set of considerations that allow us to see a possible solution to the problematic issue of Goldbach's "strong" conjecture, which amounts to asserting that any even natural number greater than 2 can be written as the sum of two prime numbers that are not necessarily distinct. Specifically, we will show mathematically that a hypothetical scenario in which no even composite number exists as a sum of two primes is impossible. This will be done by adopting a probabilistic method much simpler than the arithmetical attempts already present in literature.
Category: Number Theory
[1805] viXra:2504.0065 [pdf] replaced on 2025-04-14 00:54:42
Authors: Samuel Bonaya Buya
Comments: 20 Pages.
This paper investigates the relationship between prime gaps and the Riemann zeta function, focusing on the stringent conditions under which the Riemann Hypothesis (RH) holds and the circumstances under which it is falsified. Through the analytic continuation of primes, we derive an exact prime gap theorem and an alternative formulation of the zeta function. A key result reveals that the zeta function ��(log �������� + ����) generates infinite number of zeroes outside the critical strip. Other result reveals that a zero is generated independently of ��, providing a potential counterexample to RH. This challenges the assumption that all non-trivial zeta zeros lie on the critical line ℜ(��)=1 2 . Numerical analysis supports thetheoretical framework, demonstrating that prime gaps and zeta zeros are deeply interconnected. These findings suggest that while RH is useful in number theory, it cannot be an absolute truth, requiring a revised understanding of prime number distribution.
Category: Number Theory
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