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[11] viXra:2605.0050 [pdf] submitted on 2026-05-13 19:17:23
Authors: Jean-Yves Boulay
Comments: 6 Pages.
This work highlights a simple yet remarkably overlooked connection between the arithmetic structure underlying Sophie Germain numbers and the classical theory of triangular numbers. Although these two notions arise in distinct contexts, one in the study of prime constellations, the other in figurate number theory, they share a common algebraic backbone that becomes explicit once one examines the product x(2x + 1) arising from the transformation mapping x to (2x + 1).
Category: Number Theory
[10] viXra:2605.0046 [pdf] submitted on 2026-05-12 21:02:27
Authors: Julian Beauchamp
Comments: 4 Pages. (Note by viXra Admin: Author name is required in the article; please cite and list scientific references)
In this paper, we describe what seems to be a new Collatz-like ("if odd/if even") function, and propose some related conjectures. For any arbitrary positive number, x, iterative operations can be made such that, when even, x is divided by two, and when odd, it is added to odd integer, d. It appears that when x = 1, after sufficient iterations, the sequence always reaches 1, creating a loop. The iterative function can be stated as follows: f(x) = x/2 if x is even, x+d if x is odd.
Category: Number Theory
[9] viXra:2605.0039 [pdf] submitted on 2026-05-11 20:21:17
Authors: Christoper Mututu
Comments: 25 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We study a structural property of Goldbach representations which are expressions of even integers as sums of two primes within two specific arithmetic progressions modulo 30.We prove the following theorem by elementary modular arithmetic alone requiring no unproven hypothesis and no computation.Theorem. Let n≡8 (mod 30) with n≥38. Then every Goldbach pair (p,q) with p+q=n and p,q prime satisfies p≡q≡1 (mod 6). Furthermore, for any n≡28 (mod 30) within n≥28, every Goldbach pair (p,q) of n satisfies p≡q≡2 (mod 3) which forces both p+10 and q+10 to be divisible by 3 and therefore composite.As a consequence, no Goldbach pair of any n≡28 (mod 30) can produce a Goldbach pair of n+10 via the shift (p,q)↦(p+10,q+10).We then investigate the coupled pairs (n,n+20) where n≡8 (mod 30) observing that n+20≡28 (mod 30) always. For such a coupled pair, the shift (p,q)↦(p+10,q+10) maps a Goldbach pair of n to a Goldbach pair of n+20 automatically in terms of the sum since (p+10)+(q+10)=n+20 provided both p+10 and q+10 are prime.We define the shift-propagation count,R(n)=#{p≤n/2 ∶p prime,n-p prime,p+10 prime,n-p+10 prime}and present the following conjecture supported by extensive computation.Conjecture. For every even integer n≡8 (mod 30) with n≥38, we have R(n)≥1. That is, at least one Goldbach pair of n always shifts by +10 to produce a Goldbach pair of n+20.We verify this conjecture computationally for all 33,332 values of n≡8 (mod 30) in the range38≤n≤999,980 finding zero exceptions. The minimum value R(n)=1 occurs only at n=128 across this entire range and the average value of R(n) grows consistently with the scale of n from an average of 2.00 at the smallest values to an average of 197.69 across the full range to 10^6.We present the modular structure theorem with complete proof, state the conjecture precisely and provide full computational verification. We make no claim of proving Goldbach’s conjecture. We propose that this modular structure and the coupled pair phenomenon may serve as a foundation for future analytic work toward Goldbach’s conjecture.
Category: Number Theory
[8] viXra:2605.0038 [pdf] submitted on 2026-05-11 20:14:53
Authors: Defeng Han
Comments: 6 Pages. In Chinese (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper constructs three classes of deeply intertwined recursive sequences for Mersenne primes, encompassing all Mersenne-type core structures of the forms 2p-3, 2p-1, and 2p+3. These three classes of sequences share a common pool of prime exponents and serve as mutual recursive foundations for one another, thereby forming an organically unified recursive network. The derivations are rigorously established by relying on elementary modular arithmetic, Fermat's Little Theorem, and the $6n pm 1$ prime configuration, combined with mathematical induction and the Fundamental Theorem of Arithmetic. The terms of these sequences naturally differ by 2 from their respective "plus-two" counterparts, thereby constituting candidates for twin primes. By demonstrating the super-exponential growth property of this recursive network, the interchange of infinite quantifiers is rigorously executed within an elementary framework; this establishes the existence of a unified steady-state time and, consequently, proves the infinitude of twin primes. Simultaneously, the standard Mersenne recursive chain itself directly generates an infinite number of Mersenne primes, thereby synchronously resolving the conjecture regarding the infinitude of Mersenne primes. The entire process employs exclusively elementary number theory tools—eschewing analytic number theory and advanced sieve methods—and is logically self-consistent, free of logical gaps or leaps, and fully compliant with the axiomatic system of elementary number theory.
Category: Number Theory
[7] viXra:2605.0037 [pdf] submitted on 2026-05-11 20:08:24
Authors: J. Adnan, S. A. Dar
Comments: 9 Pages. Licensed under CC BY 4.0 (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, we investigate and evaluate the Collatz conjecture, traditionally based on positive integers, under a suitable convergence condition in which the numbers converge towards one. In our derivations, we extend the 3n+1 problem to the decimal values via a scaling factor, for showing behaviour of the last decimal digit, either even or odd. Where the numbers diverges to infinity (∞). From which it follows that between zero and one, the sequence diverges such that its limit approaches infinity.
Category: Number Theory
[6] viXra:2605.0017 [pdf] submitted on 2026-05-06 20:09:12
Authors: Edigles Bezerra Guedes
Comments: 4 Pages. (Note by viXra.org Admin: Please cite listed scientific references)
This paper determines a symmetry relation between basic hypergeometric series that has escaped the scrutiny of other mathematicians.As a direct application of this identity,we derive a double-sum symmetry and present a particular case as an exercise. Theseresults contribute to the understanding of hidden symmetries in ��-series. Moreover, it may be useful in the study of basic hypergeometricfunctions and ��-analogues of special functions.
Category: Number Theory
[5] viXra:2605.0013 [pdf] submitted on 2026-05-05 00:16:52
Authors: Song Li
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper studies the following classical geometric problem: does there exist a point inside the unit square whose distances to all four vertices are rational? We first prove that if such a point exists, its coordinates must be rational. Through a scaling transformation, the original problem is equivalently reduced to a Diophantine problem involving an integer square with integer coordinates and integer distances. Based on the parity alignment of common legs, we discuss three cases and derive contradictions using the parameterization of primitive Pythagorean triples and parity analysis. Combined with known results for boundary cases, we prove that no such point exists inside the unit square.
Category: Number Theory
[4] viXra:2605.0012 [pdf] submitted on 2026-05-04 01:46:15
Authors: Theophilus Agama
Comments: 9 Pages.
An addition chain of length h that leads to a number n is a sequence of positive integers s_0 = 1, s_1 = 2,. .. , s_h = n such that s_i = s_j + s_k (i > j ≥ k) for each 1 ≤ i ≤ h. A Brauer addition chain is the one where j = i − 1 for each 1 ≤ i ≤ h. Let l(·) and l* (·) denote the minimal length of an addition chain and the Brauer addition chain, respectively, that leads to an integer ·. Applying probabilistic methods to the iterated factor method, we show that l(2^n − 1) ≤ n − 1 + l(n) for almost all positive integers n as n −→ ∞.
Category: Number Theory
[3] viXra:2605.0011 [pdf] submitted on 2026-05-04 17:26:04
Authors: Deepak Ponvel Chermakani
Comments: 6 Pages. 7 Theorems and 2 Algorithms
Consider n runners R0, R1, ... Rn-1, with distinct constant integer speeds S0, S1, ... Sn-1 respectively, where S0=0, running around the circumference of a circle of unit circumferential length from arbitrary starting points at time t=0. At time t, denote gi(t) be the minimum absolute distance along the circumference of Ri from R0. We first use aresult on prime numbers to obtain special cases of runners speeds, for which the Lonely Runner Conjecture (LRC) is true. We then develop an approach to the LRC that derives a time at which, some subset of the runners is placed at the extremities of arcs of sectors ensuring implicit separation from R0, while the remaining runners are directly separated from R0. We use this approach to show that in the general case for large n, there exists a time T at which, gi(T) > 1/(2n) for all integers i in [1,n-1], and (g1(T) + g2(T) + ... + gn-1(T))/(n-1) tends to 1/n.
Category: Number Theory
[2] viXra:2605.0006 [pdf] submitted on 2026-05-02 23:19:56
Authors: Anthony Veglia
Comments: 7 Pages.
All higher-order hyperoperations beyond multiplication are anticommutative, featuring a pair of distinct input values being the base and the power, such as x^y. Using real whole numbers, other than the infinite trivial examples where x = y, it has been proven that 2^4 = 4^2 is the only exception to the anticommutativity property of the hyperoperation exponentiation. This proof shows that for all higher-order hyperoperations, including tetration, pentation, and beyond, thatsingular exception, H3(2, 4) = H3(4, 2), remains the sole example of "anti"-anticommutativityusing real whole number inputs.
Category: Number Theory
[1] viXra:2605.0002 [pdf] submitted on 2026-05-01 14:32:06
Authors: Muhammad Roy Asrori
Comments: 2 Pages.
In this note we give a formula for the pythagorean theorem.
Category: Number Theory
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