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[31] 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
[30] 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
[29] 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
[28] 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
[27] 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
[26] 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
[25] 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
[24] 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
[23] 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
[22] 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
[21] 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
[20] 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
[19] 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
[18] 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
[17] 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
[16] 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
[15] 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
[14] 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
[13] 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
[12] 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
[11] 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
[10] 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
[9] 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
[8] 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
[7] 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
[6] 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
[5] 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
[4] 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
[3] 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
[2] 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
[1] 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
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