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[22] viXra:2605.0116 [pdf] submitted on 2026-05-31 18:40:17
Authors: Payam Danesh, Raoul Bianchetti
Comments: 15 Pages.
This paper develops a finite framework for studying sums of prime numbers through consecutive prime gaps. The motivation comes from Ramanujan’s influence on arithmetic decompositions, partition methods, and summation ideas, while the argument remains within ordinary finite number theory. We study the partial sum of the first n primes and prove an exact identity that writes this sum as a baseline term from the initial prime together with a weighted accumulation of consecutive prime gaps. Each gap receives a weight equal to the number of later primes affected by that gap. The same identity is then expressed geometrically by encoding each weighted gap as the slope of a right triangle. Exact numerical examples verify the formula, and the asymptotic discussion shows that the decomposition has the same leading scale as the classical growth predicted by the prime number theorem. This paper also separates finite identities from regularized infinite summations. This distinction is essential: the finite formula is exact, while divergent infinite prime sums require a separate summation theory and cannot be treated as ordinary convergent series.
Category: Number Theory
[21] viXra:2605.0115 [pdf] submitted on 2026-05-31 18:39:00
Authors: J.W.L. Eerland
Comments: 4 Pages.
In an earlier paper I considered the generalized cannonball problem for r-regular polygons and studied integer solutions to the associatedDiophantine equation. In this note I prove that for every positive integer n, the triple[(r,a,b)=left(3n+2,,3n^2-2,,3n^3-3n+1ight)] is a solution. Hence the generalized cannonball problem admits infinitely many positive integer solutions. I also compare this parametric family with the 858 tuples listed in the appendix of the earlier paper. Among those tuples, 802 are generated by the present family and the remaining 56 tuples are listed explicitly.
Category: Number Theory
[20] viXra:2605.0108 [pdf] submitted on 2026-05-27 22:09:00
Authors: James DeCoste
Comments: 12 Pages.
Without any concrete feedback from the mathematical community, I was forced to have an enlightening discussion with Google's AI engine to determine what the likely problems were with my initial paper Infinitely Many Twin Primes — Proof (found on viXra.org: https://vixra.org/abs/2502.0186).Since I did all my own research in a sandbox without contributions from the mainstream mathematical community ( blind research ), I evidently ran into the same wall all prior mathematicians hit and that was proving that the twin primes candidates don't simply fizzle out as one approaches infinity. I was relying on combinatorics to prove this, which I now see was a mistake when taken by itself. As the AI engine and myself did a deeper dive on current mainstream research we noticed that much of the research I performed but did not include in my original paper were already explored by others. Some of these will be the key when reworking this on it's second take; the Hardy-Littlewood ratios 1:1:2 and the product rule that sees the pattern repeat at that product, to mention two. This was done blind without ever realizing it.We were able to determine that the rest of the original paper was sound even if not written in mainstream mathematical language. However that wall where the twin primes could simply fizzle out or completely dry up was impossible to ingore. While exploring some of the new approaches being implemented by Maynard and Tao, I made the realization that the probabilities angle should not have been abandoned. I had already done research to establish floor and ceiling limits as decaying log curves...but they too appear likely to fizzle out as one approaches infinity. That was until we took a look at a specific probability ratio of my window size sample growth versus the number of primes total growth. The first was a quadratic growth rate ( x^2) and the second was strictly linear ( number of primes grow at a steady rate). Using this probabilty along with the prior published paper finally plugs the hole and smashes down that wall as a mathematical improbability. The idea is that we have a fraction that is continually shrinking but can't actually get to absolute zero, the wall. The number of new elements (prime elimination patterns) is not increasing fast enough to keep up and eliminate all candidates in the exponentially increasing window size of my ranges. Candidates always escape. This clearly shows that my combinatoric approach was also correct in it's logic and that twin primes can never totally disappear.
Category: Number Theory
[19] viXra:2605.0104 [pdf] submitted on 2026-05-27 00:47:00
Authors: Minho Baek
Comments: 29 Pages. (Note by viXra Admin: Please cite listed scientific references)
It was already proved right that x^n+y^n=z^n, (n>2) has no solutions in positive integers which we called Fermat’s Last Theorem (FLT) by Andrew Wiles. But his proof would be impossible in the 17th century. I took the idea from Euler proof and proved in case of n=odd by simple method.
Category: Number Theory
[18] viXra:2605.0099 [pdf] submitted on 2026-05-25 16:57:08
Authors: Jose Risomar Sousa
Comments: 10 Pages.
At the negative integers, there is a simple relation between the Lerch Φ function and the polylogarithm. Starting from that relation and a formula for the polylogarithm at the negative integers known from the literature, we can deduce a simple closed formula for the Lerch Φ function at the negative integers, where the Stirling numbers of the second kind are not needed. Leveraging that finding, we also produce alternative formulae for the k-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch Φ, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we extend formulae for the Hurwitz zeta function only valid at the positive integers to the complex half-plane using this novelty.
Category: Number Theory
[17] viXra:2605.0091 [pdf] submitted on 2026-05-21 23:59:00
Authors: Ryan Hackbarth
Comments: 3 Pages. (Note by viXra Admin: Further repetition may not be accepted)
Leonard Euler considered the Zeta function for real values input of s, and in doing so he famously derived the Euler Product formulation of the Zeta function, which linked the function to prime numbers. The connection between the prime numbers and the Zeta function was further pursued by Bernhard Riemann, who constructed an analytic continuation of the function which remained valid for the entire complex plane. In doing so, he was able to link the zeroes of the Zeta function to the distribution of prime numbers. In this paper, I show that the zeroes of the Zeta Function can be identified directly from the Euler Product, and in doing so, suggest a trivial link between the primes and the zeroes of the Zeta function while bypassing the more complex machinery of analytic continuation.
Category: Number Theory
[16] viXra:2605.0073 [pdf] submitted on 2026-05-17 02:12:47
Authors: Ruiguo He
Comments: 11 Pages.
The Fransén-Robinson Constant F is closely related to e, but a discrete series representation for F has not yet been established. The lack of existing analytical identities obscures its direct algebraic analyticity, including its potential transcendence. In this paper, we introduce a novel approach utilizing the Abel-Plana Formula to derive the first discrete series representation of F, opening many possible avenues for the true nature of F.
Category: Number Theory
[15] viXra:2605.0072 [pdf] submitted on 2026-05-17 21:54:36
Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages. In French, submitted to the journal " Bulletin des Sciences Mathématiques "comments welcome.
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by: $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$ have real part} $sigma= frac{1}{2}$. In this note, I give the proof that $sigma= frac{1}{2}$ using an equivalent statement of the Riemann Hypothesis: the Dirichlet eta function.
Category: Number Theory
[14] viXra:2605.0064 [pdf] submitted on 2026-05-16 20:30:15
Authors: Julian Beauchamp
Comments: 3 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, we conjecture that a Collatz-like ("if odd/if even") function can be used to test whether a prime number has primitive root 2.
Category: Number Theory
[13] viXra:2605.0060 [pdf] submitted on 2026-05-16 20:17:43
Authors: Payam Danesh
Comments: 17 Pages. (Note by viXra Admin: For the last time, please submit article written with ANY AI assistance to ai.viXra.org!)
The Riemann Hypothesis remains one of the deepest open problems in mathematics because it connects the hidden geometry of complex zeros with the distribution of prime numbers. This paper develops a structured analytic framework that transfers the problem from its usual complex form into a real-variable setting where the desired zero structure can be studied through positivity. The central idea is to work with the completed zeta function after normalization and transformation, and to identify a precise positivity principle that would force the corresponding zeros into the correct location. The framework reduces the problem to a compact moment condition for the central logarithmic coefficients of the normalized completed function. This condition is then connected with complete Bernstein functions, Stieltjes functions, Hankel positivity, finite-difference inequalities, and an equivalent logarithmic integral representation. Each step is formulated as a rigorous implication, so the remaining difficulty is isolated in one concrete positivity theorem rather than hidden inside formal manipulation. The approach is designed to avoid the common weaknesses of many proposedarguments for the Riemann Hypothesis. It does not use the Euler product outside its valid region, does not infer the result from symmetry alone, does not replace the original zeta function with a modified object, and does not rely on numerical evidence as proof. Instead, it gives a connected chain from central derivative positivity to a Stieltjes logarithmic derivative, from there to a negative-real-axis zero structure, and finally back to the critical-line statement. The paper does not claim a completed proof of the Riemann Hypothesis. Its contribution is a clean reduction that identifies a single remaining positivity problem in a form suitable for rigorous verification. This gives a clear and testable route for future work, with finite conditions that can be studied through moment theory, operator theory, and the analytic theory of special functions.
Category: Number Theory
[12] viXra:2605.0055 [pdf] submitted on 2026-05-14 06:57:24
Authors: Bin Zhang
Comments: 8 Pages.
The Lonely Runner Conjecture states that if n runners with distinct speeds start at the same point on a unit circle, each runner will be at least 1/n away from the others at some moment. This paper provides a novel constructive analysis framework for this conjecture. We first propose the concept of a "trivial construction" — a speed configuration scheme where all non-zero runner speeds form an arithmetic progression. Using the Pigeonhole Principle, we rigorously prove that for any given threshold 1/n, this trivial construction only requires n+1 runners to ensure that the designated runner is never lonely. Furthermore, through Galilean relative transformations, this result is extended to a situation where all runners are never lonely, proving that the effectiveness of this construction remains valid after multiplying each numerator by any positive rational number. Based on this construction, we introduce a "Time-Position" geometric model, mapping the runner's motion onto a polyline on a plane. By combining features such as constant slope, speed, and the seamless splicing of the region covering the threshold curves, this model intuitively demonstrates the uniqueness and optimality of the trivial construction among all geometric configurations. It provides a solid foundation for rigorously proving that n runners must be lonely. This paper does not prove the original conjecture but provides rigorous results under specific speed configurations.
Category: Number Theory
[11] viXra:2605.0053 [pdf] submitted on 2026-05-14 16:15:18
Authors: Viktor Strohm
Comments: 8 Pages.
In accordance with the general systems theory of Yu.A. Urmantsev, systems of objects are constructed on the set of prime numbers. The relation between objects is taken as the difference of primes. Each prime number (except the first and the last) is assigned two normalized intervals: the left interval — the difference with the previous prime divided by 2, and the right interval — the difference with the next prime also divided by 2. The third coordinate (sequential number) records the multiplicity of a given combination of intervals. A three dimensional array is obtained, whose projection onto the plane (x, y) reveals a strict periodicity: in the table of residues modulo 6, cells with equal non zero residues (x mod 3, y mod 3) are empty, while the remaining cells contain only 1 or 5. A lemma is proved that explains these regularities. It is shown that the systematisation does not provide a deterministic algorithm for finding the next prime, yet it reveals deep structural constraints.
Category: Number Theory
[10] 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
[9] 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
[8] 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
[7] 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
[6] 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
[5] 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
[4] 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
[3] 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
[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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