| viXra.org > Number Theory > 2603 |
 |
 |
| viXra.org > Number Theory > 2603 |
|
|
[16] viXra:2603.0121 [pdf] replaced on 2026-03-27 10:29:52
Authors: Timothy Jones
Comments: 3 Pages. A reader gave suggestions to provide more details.
We give a proof of Pi's irrationality using geometry and some logic.
Category: Number Theory
[15] viXra:2603.0092 [pdf] submitted on 2026-03-19 01:48:42
Authors: Authman Jassim Mohammed
Comments: 3 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
A common challenge in discrete mathematics and numerical analysis is translating discrete, rule-based algorithms into continuous algebraic functions. While the foundations of this translation lie in the historic work of Joseph-Louis Lagrange, applying these principles to modern piecewise systems offers valuable geometric insights. This expository paper demonstrates an intuitive, step-by-step construction of continuous indicator polynomials to model the trivial sequence of a symmetricDiophantine equation and provides a novel polynomial alternative to Marc Chamberland’s trigonometric extension of the Collatz (3x+1) Conjecture.
Category: Number Theory
[14] viXra:2603.0090 [pdf] submitted on 2026-03-17 11:38:20
Authors: Richard J. Mathar
Comments: 13 Pages. Reprinted from a version dated from 2012.
A Pisano period is the period of an integer sequence which is obtained by reducing each term of a primary sequence modulo some integer m >= 1.Each primary sequence which obeys a linear homogeneous recurrence with constant coefficients has such periods for all m >= 1. The manuscript tabulates the lengths of these periods indexed by m for a small subset of the OEIS sequences.
Category: Number Theory
[13] viXra:2603.0071 [pdf] submitted on 2026-03-13 17:37:32
Authors: Francesco Aquilante
Comments: 3 Pages.
The concept of the textit{quantum Riemann sum} ($Q$-sum) is introduced as a theoretical framework to bridge the gap between the physics of discrete energy quantization and the analytic continuation of divergent series. By identifying the $Q$-sum generator as a Todd operator, we demonstrate that the Riemann zeta function emerges as the spectral signature of a complex dynamical system. We show % derive a closed-form energy identity revealing that the non-trivial zeros $ho=sigma+igamma$ correspond to states of vanishing boundary flux, where the system’s hermitian potential and anti-hermitian flow reach perfect equilibrium,for which charge-parity symmetry % governed by charge-parity symmetry, that is conserved if and only if requires the states to reside on the critical line $sigma = 1/2$. Consequently, the {em Riemann Hypothesis} of 1859 is now revealed not merely as a proven arithmetic theorem, but as a necessary condition for spectral stability and symmetry conservation in quantized systems.
Category: Number Theory
[12] viXra:2603.0062 [pdf] submitted on 2026-03-11 20:50:18
Authors: Walter A. Kehowski
Comments: 135 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
If the digits of number n (in base b) can be split p|q so that n=rq rp$, where rp and rq are the numbers formed by reversing the digits of p and q, respectively, then n is called a phoenix number in base b. If q has k digits in base b, then n is called a k-Phoenix number in base b. For example, 1.1.n.n|b is 1-Phoenix in base b=n^2-1. The existence of infinitely many k-phoenix numbers in any base b is claimed, but not proven.
Category: Number Theory
[11] viXra:2603.0057 [pdf] replaced on 2026-03-20 09:16:45
Authors: Timothy Jones
Comments: 5 Pages. A reader provided some corrections.
We give a proof of pi's irrationality that references principles of set theory and cardinality within the context of basic geometric properties of a circle.
Category: Number Theory
[10] viXra:2603.0051 [pdf] replaced on 2026-03-12 13:29:02
Authors: Samuel Datu Castelo
Comments: 3 Pages.
This paper presents a derivation of the closed-form expression for the alternatingpower sum without employing Euler polynomials, which are traditionally used in suchformulations. Instead, the derivation utilizes the shifted series method together withdifferentiation techniques to construct a polynomial representation for the sum. Theresulting expression is then connected to classical number-theoretic constants throughthe Riemann zeta function and Bernoulli numbers. This approach provides an alterna-tive and more elementary framework for obtaining the alternating power sum formulawhile avoiding the formal machinery of Euler polynomials.
Category: Number Theory
[9] viXra:2603.0050 [pdf] submitted on 2026-03-08 22:11:15
Authors: Christoper Mututu
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific references)
We introduce and rigorously define a new class of prime numbers, which we term as pure primes. A natural number is classified as a pure prime if, for every contiguous partition of its decimal representation into equal length segments, each segment is itself a prime number. Restricting the digits to the prime set {2, 3, 5, 7}, we enumerate pure primes across all digit lengths. Specifically, our results establish the following counts: There are 4 total 1-digit pure primes. There are 4 total 2-digit pure primes. There are 15 total 3-digit pure primes. There is only 1 4-digit pure prime. There are 128 total 5-digit pure primes. There are 0 total 6-digit pure primes. There are 1325 total 7-digit pure primes. There are 0 total 8-digit pure primes. There are 469 total 9-digit pure primes. There are 0 total 10-digit pure primes. There are 214432 total 11-digit pure primes. There are 0 total 12-digit pure primes. There are 2884201 total 13-digit pure primes. There are 10 total 14-digit pure primes. There are 236 total 15-digit pure primes. As a result, the total pure primes across 1-15 digits are 3100825. These numbers span a huge range from 1-digit primes 2, 3, 5, 7 up to 15-digit primes in the quadrillion fully enumerated by our exhaustive computation. While a formal proof of absolute finiteness remains open, this extensive enumeration demonstrates that within this range, the class of pure primes is tightly constrained and structurally self-limiting. The partition invariance requirement imposes increasingly restrictive combinatorial conditions as the digit length grows. Empirically, this leads to long stretches of digit lengths admitting no pure primes. While these observations do not constitute a proof of finiteness, they indicate strong structural sparsity within this digit constrained prime class. This could hint at deep links between digit partition invariance and finiteness properties analogous to those conjectured in other constrained number sets. Our exhaustive computational investigation reveals strong structural constraints. Pure primes exist only for certain digit lengths with multiple lengths producing no examples. Beyond its combinatorial elegance, the discovery of pure primes opens new avenues for research into digit partition invariance, prime density constraints and the structure of prime subsets in discrete number spaces. These findings suggest a previously unrecognized form of order in the prime landscape, providing both a novel mathematical object and a framework for exploring finiteness within prime number theory.
Category: Number Theory
[8] viXra:2603.0046 [pdf] submitted on 2026-03-08 21:50:57
Authors: Leckan M. Sibanda
Comments: 16 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
We present a complete, rigorous proof of the Riemann Hypothesis for the Riemann zeta function and its generalization to all primitive Dirichlet L-functions. The proof is based on a fundamental property of the zeros: for each zero (the imaginary part of a nontrivial zero) there exists a positive integer (called the optimal modulus) such that the zero times this integer divided by pi is exceptionally close to an integer. This Diophantine approximation leads to an exact phase-locking recurrence derived from the logarithmic identity. Using a decomposition of the Dirichlet series into residue classes, we obtain representations of the zeta function and its symmetric counterpart in terms of real, positive, strictly decreasing amplitudes and a fixed set of roots of unity. The vanishing of a certain anti-symmetric combination of the zeta function at a zero forces a simple trigonometric condition whose only solution is that the real part equals one-half. The argument is elementary, self-contained, and extends naturally to all Dirichlet L-functions. Numerical verification confirms the existence of the optimal modulus and the phase-locking identity to extraordinary precision, but the proof itself is purely analytic.
Category: Number Theory
[7] viXra:2603.0041 [pdf] submitted on 2026-03-07 21:52:20
Authors: Youssef Ayyad
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This article establishes a fundamental connection between number representation in different bases and the geometry of regular polygons. We demonstrate that every positive integer $N$ admits a unique decomposition $N = b^m - R$ where $b^m$ is the smallest power of the base $b$ exceeding $N$, and $R$ is expressed in base $b$. This arithmetic fact translates geometrically into an angular homothety mapping the regular $b^m$-gon to the $N$-gon. Through this geometric lens, we obtain a natural classification of numbers: primes appear as elements whose associated fractions are irreducible regardless of the base. Our main result is a striking geometric characterization of twin primes: for a prime $p$, the pair $(p, p+2)$ consists of twin primes if and only if for every base $b$ with $2 leq b < p$, the ratio of homothety factors $lambda_b(p+2)/lambda_b(p)$ equals $p/(p+2)$. We explore connections between this characterization and the twin prime conjecture, offering a new geometric perspective on this ancient problem.
Category: Number Theory
[6] viXra:2603.0035 [pdf] replaced on 2026-04-06 01:34:28
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
[5] 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
[4] viXra:2603.0021 [pdf] replaced on 2026-05-08 11:15:40
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
[3] 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
[2] 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
[1] 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
| Third-Party Links: |
|