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[33] viXra:2412.0188 [pdf] replaced on 2026-02-27 21:43:44
Authors: Toshihiko Ishiwata
Comments: 10 Pages.
This paper is a trial to prove Goldbach conjecture according to the following process.1. We make the function lu2032(n) regarding n. When an even number n is divided into 2 odd numbers, lu2032(n) ≤ l(n) holds true. l(n) : the total number of ways to divide an even number n into 2 primenumbers. 2. We find that 0 < lu2032(n) holds true in 4∗1018 ≤ n. 3. Goldbach conjecture is already confirmed to be true up to n = 4∗1018. 4. Goldbach conjecture is true from the above item 1. — 3.
Category: Number Theory
[32] viXra:2412.0186 [pdf] submitted on 2024-12-30 10:45:17
Authors: Hans Hermann Otto
Comments: 7 Pages.
We present continued fraction representations of universal numbers and some approximations to underline on what nature’s infinitely repeated processes are footed as the mathematical basis of our life and the entire cosmos.
Category: Number Theory
[31] viXra:2412.0174 [pdf] submitted on 2024-12-28 23:28:55
Authors: Berkouk Mohamed
Comments: 22 Pages. In French
The Collatz conjecture which is defined by: starting from all positive integer n, we apply these 2 instructions, if it is even divide it by 2, and if it is odd we multiply it by 3, we add 1 and we divide all by 2, the conjecture states that if we repeat the transformation of n, we always end up falling on 1 which in turn finds itself trapped indefinitely in the trivial cycle 1, 2, 3, 4. three steps were taken to arrive at its demonstrations, the first led to the uniqueness of the Trivial Cycle which consisted, in PART 1, of exploring in the Trivial Cycle a property recurring in all other cycles with more than three terms of which they can only generate, according to their Cycle equation, positive integers 1, 2 or 4. The second step consists of constructing a direct equivalence between the convergence of Syracuse towards 1 with the fact that the ratio of odd terms to the number of even terms is stuck in the interval [0, 0.63092975..[, and that thanks to this proposition, we were able to demonstrate the decrease of all n towards 1, finally the third step, which by a property of the average demonstrated by recurrence the equation which connects the powers of 2 with the powers of odd terms to which we applied the —Collatz 3n +1 instruction.
Category: Number Theory
[30] viXra:2412.0169 [pdf] replaced on 2025-01-01 22:08:13
Authors: Algoni Mohamed
Comments: 10 Pages.
We propose a method to solve the Diophantine equation (E)1 : ax + by = c, where a, b, andc are natural numbers with a, b ̸= 0 and a≡±(10/p)[b], where p ∈ {1, 2, 5, 10}. The method is based on analyzing the units of the products pc and pb, denoted as u and uu2032, respectively. This approach simplifies and accelerates the resolution process by focusing on these units.
Category: Number Theory
[29] viXra:2412.0165 [pdf] submitted on 2024-12-26 23:27:30
Authors: Zhiyang Zhang
Comments: 32 Pages.
The Riemann hypothesis was proposed by mathematician Bernhard Riemann in 1859. The usual view is that Riemann makes a guess that all non trivial zeros are located on the critical line after simply calculating a few zeros, but this is not the case. Riemann even knew that the probability of counterexamples occurring was very low at that time, so he shyly put forward such a hypothesis. This article provides a detailed explanation of the conditions for the existence of counterexamples in Riemann's hypothesis from his perspective, and provides a calculation method for counterexamples.
Category: Number Theory
[28] viXra:2412.0154 [pdf] submitted on 2024-12-25 00:55:35
Authors: Takashi Nakajima
Comments: 18 Pages. (Version histories/notes at impropriate place blocked by viXra Admin - Please cite and list scientific references)
If s=a+bi, proving that the absolute value of the imaginary part of ξ(s)=ζ(s)Γ(s/2)π^(-s/2) is not 0 for 0<a<0.5, 0.5<a<1, -∞<b<∞ is equivalent to proving that the Riemann hypothesis is true. When searching for 0<b<x (x is near 0) for 0<a<0.5, 0.5<a<1 using a computer, it was confirmed that the absolute value of the integral term of |Im(ξ(s))| is smaller than the absolute value of the additional term of |Im(ξ(s))| and that the polarities of the two are the same. It was shown that even when b→∞, the absolute value of the integral term of |Im(ξ(s))| remains smaller than the absolute value of the additional term of |Im(ξ(s))| and converges. The polarities of the two are the same. As a result, the Riemann hypothesis was proven correct.
Category: Number Theory
[27] viXra:2412.0150 [pdf] submitted on 2024-12-24 01:35:29
Authors: Lynette Michael Winslow
Comments: 70 Pages.
We present unconditional proofs of the convergence of reciprocal sums associated with certain special prime sequences defined by polynomial and multiplicative conditions. In particular, consider a polynomialP(x) = sum_{i=0}^{n} c_i x^i quad text{with integer coefficients and } c_n > 0.Define S_P = { p : p text{ prime and } P(p) text{ is prime} }. We prove thatsum_{p in S_P} frac{1}{p} < infty. Moreover, we establish explicit upper and lower bounds for both the counting functionpi_P(x) = |{ p in S_P : p le x }| and the partial sums sum_{substack{p in S_P p le x}} frac{1}{p}. Next, we consider Balanced Primes, defined by the condition that each balanced prime p_n forms a three-term arithmetic progression with its neighbors: p_n = frac{p_{n-1} + p_{n+1}}{2}. Applying our multi-level sieve methods to these primes, we similarly prove the convergence of their reciprocal sum and provide corresponding quantitative estimates. In addition, we examine the set of emph{Good Primes}, defined by the multiplicative inequality p_n^2 > p_{n-i} cdot p_{n+i} quad text{for all } 1 le i le n-1.We show that their reciprocal sum also converges and provide corresponding upper and lower bounds on their counting functions and partial reciprocal sums. Our approach, which we call the M-Brun Sieve, refines classical sieve methods into a multi-level framework that can handle intricate polynomial and multiplicative constraints simultaneously. Notably, our results do not rely on any unproven conjectures. These findings yield substantial new insights into the distribution and density of these special classes of primes, thereby resolving longstanding questions posed by Pomerance regarding Good Primes.
Category: Number Theory
[26] viXra:2412.0148 [pdf] submitted on 2024-12-24 01:30:36
Authors: Janko Kokosar
Comments: 5 Pages.
The paper presents two procedures, or derivations with educated guessing, where Ramanujan’s formula for $pi^4$ is the final result. The first derivation uses the integer approximation for $pi^3$, and 22/7 as an approximation for $pi$, followed by a simple modification to match Ramanujan’s formula. The second derivation uses an integer approximation for $pi^5$ alongside with 22/7. Once again, the modification to Ramanujan’s formula for $pi^4$ is quite simple. The aim is also to explore whether further approximations of this formula exist. Some hints are provided on this subject.
Category: Number Theory
[25] viXra:2412.0147 [pdf] submitted on 2024-12-23 20:47:24
Authors: Daoudi Rédoane
Comments: 4 Pages.
In this article, we introduce a new mathematical conjecture that connects the sum of divisors of integers, prime-counting functions, and the properties of prime numbers. The conjecture incorporates basic number-theoretic functions, including the sum of divisors of an integer n, the sum of the squares of the divisors of n and the prime-counting function π(n), which counts the number of prime numbers less than or equal to n
Category: Number Theory
[24] viXra:2412.0143 [pdf] submitted on 2024-12-23 01:55:13
Authors: Theophilus Agama
Comments: 4 Pages.
We provide an elementary proof of Fermat's last theorem using the notion of olloids.
Category: Number Theory
[23] viXra:2412.0139 [pdf] submitted on 2024-12-22 15:45:42
Authors: Timothy Jones
Comments: 3 Pages.
We prove the contrapositive of pi is rational implies lim sin n! as n goes to infinity converges to show pi is irrational.
Category: Number Theory
[22] viXra:2412.0136 [pdf] submitted on 2024-12-21 15:04:58
Authors: ChaeWon Hwang
Comments: 2 Pages. (Note by viXra Admin: Please cite and list scientific references)
I will prove that Collatz's hypothesis is true.
Category: Number Theory
[21] viXra:2412.0132 [pdf] submitted on 2024-12-22 03:27:51
Authors: Israr Luqman
Comments: 5 Pages. (Note by viXra Admin: Please submit article in pdf format only)
This paper proposes the Octa-Quadrant System (OCS), an innovative extension of the Cartesian coordinate system that divides the plane into eight quadrants instead of the traditional four. Each quadrant spans 45° and is grouped into unique pairs to form a novel coordinate framework. This extension allows new algebraic operations, geometric interpretations, and rotational symmetries to be explored. Theoretical foundations, algebraic properties, and potential applications of the OCS in mathematics, physics, and engineering are discussed, alongside a comparison with existing systems and their limitations.
Category: Number Theory
[20] viXra:2412.0130 [pdf] submitted on 2024-12-22 03:09:00
Authors: Edgar Valdebenito
Comments: 3 Pages.
In this note we give some formulas related to Pi.
Category: Number Theory
[19] viXra:2412.0116 [pdf] submitted on 2024-12-19 12:35:13
Authors: Timothy Jones
Comments: 7 Pages.
It is shown that the limit of cos(j) and sin(j) as j goes to infinity does not exist. DeMoivre's theorem implies cos j! + i sin j! raised to the 1/(j-1)! power equals cos j + i sin j. Assuming pi is rational, its multiple can be expressed as a factorial. This implies that cos(j) and sin(j) converges, a contradiction.
Category: Number Theory
[18] viXra:2412.0093 [pdf] submitted on 2024-12-17 03:37:28
Authors: Casey Allard
Comments: 10 Pages.
The ‘Twin Prime Conjecture’ posits that there are infinitely many pairs of prime numbers separated by a gap of exactly two (p, p+2). This proof uses the concepts of modular residues, gaps ("12p + 36" pattern), and digit sums. The proof integrates concepts of gap growth, residue non-exhaustion, and digit sum cycles, ensuring twin primes persist infinitely.
Category: Number Theory
[17] viXra:2412.0085 [pdf] replaced on 2024-12-18 04:58:58
Authors: John Yuk Ching Ting
Comments: 23 Pages. Generalized Riemann hypothesis and Generalized BSD conjecture
This expository-styled paper contains 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 Analytic rank and Symmetry type of L-functions. On the Sign normalization when applied to eligible L-functions, we posit its dependency on even-versus-odd Analytic ranks, degree of L-function, and the particular gamma factor present in functional equations for Genus 1 elliptic curves and higher Genus curves. The relevant mathematical arguments are postulated to satisfy Generalized Riemann hypothesis, and Generalized Birch and Swinnerton-Dyer conjecture. We explicitly mention their underlying proven/unproven hypotheses or conjectures.
Category: Number Theory
[16] viXra:2412.0078 [pdf] replaced on 2025-10-15 20:49:24
Authors: Francesco Aquilante
Comments: 4 Pages.
It is shown that if a triple of distinct positive integers $(a,b,c)$ were to exist such that $a^n+b^n=c^n$ for some odd integer $ngeq 3$, then it must be Pythagorean, i.e. $a^2+b^2=c^2$ must hold too, from which a contradiction arises since this is possible only if either $a$ or $b$ are zero. We arrive at this conclusion by investigating the (partial) trace of a model hamiltonian operator whose energy levels correspond to those of the so-called H"uckel hamiltonian applied to rings containing an odd number of atomslying on a M"obius strip rather than a planar topology. Furthermore, the contradictory nature of our result implies the correctness of the associated statement contained in the famous Fermat's Last Theorem. Given the use of concepts from quantum mechanics and matrix algebra unknown at his time, and the fact that the essence of the present proof may not fit within a margin of a typical book, mystery still remains over Pierre de Fermat's {em demonstrationem mirabilem}.
Category: Number Theory
[15] viXra:2412.0076 [pdf] replaced on 2025-08-21 17:54:14
Authors: Barry Foster
Comments: 1 Page.
Fermat's Last Theorem (FLT) states that there are no natural numbers A, B, C, and n such that A^n = B^n + C^n is true for n>2.The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica:https://en.wikipedia.org/wiki/ArithmeticaFermat added that he had a proof that was too large to fit in the margin and because he had done likewise for other since-proved theorems there has since been a search to find a short proof.This effort examines the attributes of the numbers in FLT and shows that irrationalnumbers are required for it to be true
Category: Number Theory
[14] viXra:2412.0047 [pdf] submitted on 2024-12-08 21:40:55
Authors: Juan Elias Millas Vera
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references)
I show in this paper some relations of functions which following the logic of the definition of all of them, it shows a concise perspective of the function π(n), namely the number of primes less than a number n.
Category: Number Theory
[13] viXra:2412.0046 [pdf] submitted on 2024-12-08 21:40:23
Authors: Juan Elias Millas Vera
Comments: 1 Page. (Note by viXra Admin: Please cite and list scientific references)
In this short paper I enunce a computational method to obtain with any natural number n the n-th prime number without use Riemann Hypotesis.
Category: Number Theory
[12] viXra:2412.0045 [pdf] submitted on 2024-12-08 15:14:10
Authors: Joseph Musonda
Comments: 6 Pages.
In 1742, a German mathematician Christian Goldbach proposed a Goldbach conjecture. The conjecture states that every even integer greater than 2 can be expressed as a sum of two prime numbers. The conjecture has been verified up to 4 000 000 000 000 000 000 and no counter example has been given up to date. Before proving this conjecture, a prime generating function was created which relates to triangular numbers. The triangular numbers are categorized into special Vm and non- special M triangular numbers. The special triangular numbers Vm are used to generate all prime numbers (pm) greater than 2 (0.375 is included among Vm to generate 2) using the function Pm=√(8Vm + 1). The Vm is obtained from Tn∩Mu2032. The proof holds true for all even integers greater than 2. The proof is so important in number theory because it involves a prime number generating function. This function may help us solve many prime number related problems. Keywords: Triangular numbers, special, non-special, Goldbach conjecture, prime numbers
Category: Number Theory
[11] viXra:2412.0036 [pdf] submitted on 2024-12-06 21:52:18
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 35 Pages.
The Critical Symmetry Theorem transforms number theory by embedding prime distributions within deterministic harmonic periodicities. By enforcing symmetry, it aligns all non-trivial zeros of the Riemann zeta function (zeta(s)) on the critical line (Re)(s) = (0.5), resolving the Riemann Hypothesis. The theorem unifies key conjectures, including Twin Primes, Goldbach’s Conjecture, and bounded prime gaps, as natural consequences of symmetry. This framework bridges chaos and order, reshaping number theory into a deterministic system governed by harmonic principles.
Category: Number Theory
[10] viXra:2412.0026 [pdf] submitted on 2024-12-05 02:44:34
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 2 Pages.
This document refines the time-based suppression framework, unifying it with the truths established in the Singular Proof. The suppression function ( k(x) ) provides stability for residual terms derived from zeta zeros, ensuring the absence of off-critical zeros and supporting the Riemann Hypothesis. This refinement solidifies the Bourgeois Prime Distribution Model as a universal framework for prime behavior and harmonic alignment.
Category: Number Theory
[9] viXra:2412.0025 [pdf] submitted on 2024-12-05 02:44:19
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 3 Pages.
The Refined Prime Gaps Conjecture posits that non-trivial zeros of the Riemann zeta function contribute periodic corrections to prime gaps, refining their distribution. This document provides empirical validation of the conjecture using data from the first 10,000 primes, formalizes residual bounds, and connects periodic corrections to the conjecture’s theoretical framework. Additionally, this document explores a 1-to-1 relationship between the conjecture’s validity and structured periodicity in prime gaps.end{abstract}
Category: Number Theory
[8] viXra:2412.0024 [pdf] submitted on 2024-12-05 02:44:04
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 2 Pages.
We conjecture that the residual bounds of the prime-counting function (pi(x)) exhibit logarithmic decay, given by:[|G(x)| leq frac{k}{ln x}, ]where (G(x) = pi(x) - text{Li}(x)) represents the residual difference between the prime-counting function (pi(x)) and the logarithmic integral (text{Li}(x)), and (k) is the universal scaling constant with an empirically validated value of (k = 1.23 pm 0.05). This conjecture is underpinned by the interplay between prime gaps, residual suppression, and the periodic contributions of zeta zeros. Testing residual bounds up to (x = 10,000,000) confirms logarithmic decay with no deviation from the expected suppression formula.
Category: Number Theory
[7] viXra:2412.0023 [pdf] submitted on 2024-12-05 02:55:50
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 2 Pages.
This document formalizes the conjecture that the periodic correction term: [cos(2pi ho log x),]where (ho) represents the imaginary components of the non-trivial zeros of (zeta(s)), universally aligns with residual suppression across all (x > 1). This alignment stabilizes residual bounds and reinforces the logarithmic decay of (|G(x)|). Theoretical justification and empirical evidence are provided to support this conjecture, with a focus on critical line symmetry and logarithmic modulation.
Category: Number Theory
[6] viXra:2412.0018 [pdf] submitted on 2024-12-05 08:47:43
Authors: Wladislaw Zlatjkovic Petrovescu
Comments: 2 Pages.
In this paper we refute M. Detic.
Category: Number Theory
[5] viXra:2412.0016 [pdf] replaced on 2025-04-07 15:57:56
Authors: Maurizio M. D'Eliseo
Comments: 4 Pages.
Power sums can be reinterpreted as weighted sums of odd sequences using a simple transformationof Riemann sums into Lebesgue sums.This reformulation introduces a self-referential recursive framework in which power sums are expressed as linear combinations of power sums in descending order.
Category: Number Theory
[4] viXra:2412.0012 [pdf] submitted on 2024-12-04 22:24:12
Authors: Blaize Rouyea, Corey Bourgeois, Trey Bourgeois
Comments: 7 Pages.
The Riemann Hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function (zeta(s)) satisfy (text{Re}(s) = 0.5). This conjecture, bridging the chaotic behavior of primes with the structured elegance of analytic functions, has stood as a cornerstone of mathematical inquiry for over a century. In this work, we introduce the Rouyea-Bourgeois Prime Model (RBPM), a groundbreaking framework that decodes prime behavior and redefines our understanding of prime-driven harmonics in relation to zeta zeros.vspace{1em}By rethinking the prime counting function (pi(x)), we identify the symmetry function:[S(s) = sum_{p text{ prime}} frac{1}{log(p)} p^{-s},]which encapsulates the harmonic contributions of primes. Through destructive interference of oscillatory terms, (S(s)) enforces the critical line alignment of all non-trivial zeros:[F_{text{total}}(t) = F_{text{prime}}(t) + F_{text{composite}}(t) = 0 quad implies quad text{Re}(s) = 0.5.]The functional equation of (zeta(s)) further guarantees symmetry across the critical strip, confirming RH as an inevitable outcome of harmonic symmetry. This proof unites chaos and order, bridging primes and analytic functions to inaugurate a new era of harmonic mathematics. The implications extend far beyond the resolution of RH, opening pathways to innovative discoveries across number theory and analytic frameworks.
Category: Number Theory
[3] viXra:2412.0006 [pdf] replaced on 2024-12-05 19:36:49
Authors: Bassam Abdul-Baki
Comments: 4 Pages.
On real numbers whose fractional part remain constant after squaring.
Category: Number Theory
[2] viXra:2412.0004 [pdf] submitted on 2024-12-01 15:55:45
Authors: Berndt Gensel, Theophilus Agama
Comments: 16 Pages.
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the expansion principles as central tools for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.
Category: Number Theory
[1] viXra:2412.0003 [pdf] replaced on 2025-08-30 10:31:05
Authors: Mar Detic
Comments: 3 Pages.
This paper describes a deterministic primality test based on thedivisibility properties of binomial coefficients modulo a candidate integer n. The test checks the condition nk≡ 0 (mod n) for all k inthe range 1 ≤ k ≤ ⌊log2 n⌋. This criterion is a direct consequence ofthe polynomial identity (x + 1)n ≡ xn + 1 (mod n), which holds ifand only if n is prime. The algorithm uses an efficient recurrence relation, achieves a time complexity of Oe(log3 n), and correctly identifiescomposite numbers, including Carmichael numbers.
Category: Number Theory
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