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[17] viXra:2604.0104 [pdf] submitted on 2026-04-27 17:48:41
Authors: Abdelmajid Ben Hadj Salem
Comments: 117 Pages.
This booklet is the tome IV of my Selected Papers. It is my mathematical contributions in the field of the Number Theory during the last 12 years.
Category: Number Theory
[16] viXra:2604.0097 [pdf] submitted on 2026-04-26 18:30:17
Authors: Yuhua Li
Comments: 12 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
In this paper,we proved that for classic integral representation of Riemann $xi$-function $xi(s)=frac{1}{2}+frac{s(s-1)}{2}int_1^inftypsi(x)(x^{s/2-1}+x^{(1-s)/2-1})dx=-4psi'(1)+int_1^inftypsi'(x)((1-s)x^{s/2}+sx^{(1-)/2})dx=2int_1^infty(frac{3}{2}psi'(x)+xpsi''(x))(x^{s/2}+x^{(1-s)/2})dx$ , the common lower limitation 1 of the three divergent series equals each other (including $frac{-1}{2}$ for the first and $(-4)psi'(1)$ for the second).This provides a new approach to prove the three integral representation without using partial integration.
Category: Number Theory
[15] viXra:2604.0085 [pdf] submitted on 2026-04-22 20:33:08
Authors: Timothy Jones
Comments: 2 Pages. (Note by viXra Admin: Please cite listed scientific references)
The point on a unit circle that is associated with the arc Pi/2 is (0,1). We prove the bidirectional: every ordered pair of positive real numbers (a,b) corresponds to a point on the circumference of a circle of radius square root of a^2+b^2 with an associated radius with an arc length less than Pi/2. If Pi/2 is rational this gives a contradiction.
Category: Number Theory
[14] viXra:2604.0079 [pdf] submitted on 2026-04-21 23:49:01
Authors: Payam Danesh
Comments: 19 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
A Gamma—Bernoulli approach to the critical-line problem for the Riemann zeta function is developed. Starting from the Mellin—theta representation and the functional equation, one obtains explicit identities for the reflected Gamma quotient and for the regularization built into the Weierstrass product for Γ. On the Bernoulli side, the kernel (eu − 1)−1 is decomposed into its singular part and an analytic remainder, which yields a concrete zero-conditioned identity after continuation. The analysis shows that the harmonic divergence visible in raw finite Gamma products is a truncation phenomenon and therefore cannot by itself force Re(ρ) = 1/2. What remains is a coercive estimate which, if established, would convert the same mechanism into a critical-line theorem.
Category: Number Theory
[13] viXra:2604.0071 [pdf] submitted on 2026-04-19 18:58:09
Authors: Jose Risomar Sousa
Comments: 12 Pages.
A generalization of the Riemann functional equation with a broader validity domain than the one available in the literature is introduced. The insight that led to this new relation came from a new formula for the zeta function created herein that implies the Riemann functional equation. A few minor developments that stem from new formulae introduced previously are also discussed.
Category: Number Theory
[12] viXra:2604.0068 [pdf] submitted on 2026-04-18 23:35:24
Authors: Rusin Danilo Olegovich
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We introduce and study the entire function $lambda(s) = sum_{n=1}^{infty} n^{s+1/2}/n^n$,defined by a Dirichlet-type series with super-exponential coefficients. We prove that$lambda(s)$ converges absolutely for all $s in mathbb{C}$, uniformly on compact sets,and is therefore an entire function of order zero. We establish a closed-form evaluationof the special value $lambda(-1/2) = int_0^1 x^{-x}, dx$, connecting $lambda$ to theclassical Sophomore's Dream identity of Bernoulli. We further prove that $lambda(s)$ isterm-by-term differentiable, with $lambda^{(k)}(s) = sum_{n=1}^{infty} (ln n)^k n^{s+1/2}/n^n$for all $k geq 0$, justified by the Weierstrass $M$-test. Finally, we propose a conjectureconnecting $lambda(s)$ to the Riemann zeta function.
Category: Number Theory
[11] viXra:2604.0063 [pdf] submitted on 2026-04-17 17:15:56
Authors: Ryan Hackbarth
Comments: 4 Pages.
In this paper, I demonstrate how the Dirichlet Eta function may be represented as a sum of its fundamental frequencies through the use of a power series whose coefficients are the Euler Product representation of the Riemann Zeta function.
Category: Number Theory
[10] viXra:2604.0061 [pdf] submitted on 2026-04-17 00:26:11
Authors: Xiaofeng Hu
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
The Collatz conjecture states that for any given positive integer N,if N is even, divide it by 2; if N is odd, multiply it by 3 and add 1.Repeating this process,N will eventually become 1. This paper provesthat any positive odd integer O other than 1 cannot return to itself nomatter how many times the iteration is performed. We derive the generalformula satisfying this condition and rigorously prove by mathematicalinduction that this formula equals 1 uniquely in the set of positive oddintegers. We thus conclude that there are no non-trivial periodic orbitsin the Collatz mapping.
Category: Number Theory
[9] viXra:2604.0052 [pdf] submitted on 2026-04-15 19:40:31
Authors: S. Mayank
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references)
This paper presents a novel iterative representation of Fermat numbers, defined by the sequence Fn = 2^(2^n) + 1. By leveraging the fundamental recurrence relation F(n+1) - 2 = Fn(Fn - 2), we define a functional equation x = A/x - 2, where A = F(n+1) - 2. We demonstrate that this equation yields two integer solutions, x1 = Fn - 2 and x2 = -Fn. Through an analysis of the derivative of the map f(x) = A/x - 2, we prove that x = -Fn is an attractive fixed point and x = Fn - 2 is a repulsive fixed point, leading to a unique, convergent infinite continued fraction for the negative of any Fermat number. This provides a bridge between the rapid growth of Fermat sequences and the stability of iterative rational functions.
Category: Number Theory
[8] viXra:2604.0049 [pdf] submitted on 2026-04-14 20:24:27
Authors: Christoper Mututu
Comments: 13 Pages. (Note by viXra Admin: For the last time, please cite and list scientific references)
We introduce a deterministic construction for generating composite number pairs (A,B) from strictly isolated prime sextets, configurations of exactly six primes situated at fixed offsets {0,8,14,18,24,32} from a base value a≡9 (mod 10) with no additional prime existing anywhere within the interval [a,a+32]. We term such configurations strictly isolated prime constellations of order six.The structural constraint a≡9 (mod 10) forces the six primes to terminate in the digit pattern 9,7,3,7,3,1 respectively which is a consequence of the fixed offsets modulo 10. These six primes are arranged into a 2×4 rectangle whose columns are indexed by the digits {1,3,7,9}, the complete set of possible terminal digits of any prime greater than 5. Column wise addition and subtraction yield a Sums row and a Difference row from which the composite A and B are defined by their respective totals.We prove that this construction satisfies four universal invariants. First, the closed form identities A=6a+96 and B=2a+52 hold for every valid cluster. Second, A is always divisible by 1,2,3,5 and 6 while B is always divisible by 1,2,5 and 10, both following algebraically from a≡9 (mod 10). Third, the decimal expansion of A/B always carries a signature 2.9u2026 and B/A always carries initial signature 0.3u2026 for all a≥274 proven via closed form analysis. Fourth, both A/B and B/A always produce non terminating repeating decimal expansions guaranteed by the arithmetic structures of the reduced denominators.These four invariants are established algebraically and confirmed computationally across 17,138 valid clusters up to 100 billion with zero failures on every claim.
Category: Number Theory
[7] viXra:2604.0048 [pdf] submitted on 2026-04-14 19:25:48
Authors: Amal Ladjeroud
Comments: 10 Pages.
Hilbert-Pólya conjecture is proved by constructing Hilbert-Pólya operator, the self-adjoint operatorwhere its eigenvalues are exactly the imaginary parts of zeros of Riemann zeta function on the critical line. Hence, the Riemann hypothesis is true.
Category: Number Theory
[6] viXra:2604.0047 [pdf] submitted on 2026-04-13 20:43:28
Authors: Pranshu Tripathi
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references)
The collatz conjecture was introduced by Lothar collatz in 1937. It is also known as "3n + 1 problem". The conjecture states: Start from any positive integer, n. If n is even, divide by 2; if n is odd, multiply by 3 and add 1. Now, the conjecture says that if you keep repeating above steps, you will finally reach 1, no matter what value of n we choose. In this paper, we prove collatz conjecture using method of mathematical induction. We also use binary and trenary numbers to prove collatz conjecture.
Category: Number Theory
[5] viXra:2604.0042 [pdf] submitted on 2026-04-11 21:48:39
Authors: Kenneth A. Watanabe
Comments: 19 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a formal proof of Brocard’s Conjecture, which posits that there are at least four prime numbers between the squares of any two consecutive primes pi2 and p_i+12 for i>1. By defining the function π*(n) that approximates the prime counting function π(n), we establish a lower bound for the number of primes in these intervals. Using mathematical induction, we demonstrate that the minimum number of primes in the interval, Δπ*(pi), is consistently greater than or equal to 4 for all pi ≥ 3. The proof is further supported by a rigorous error analysis, bounding the maximum possible deviation between the estimated prime count π*(n) and the actual prime counting function π(n).
Category: Number Theory
[4] viXra:2604.0035 [pdf] submitted on 2026-04-11 01:25:28
Authors: Idan Hackmon
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We prove that no covering system with distinct odd moduli can have its least common multiple supported on at most four distinct odd primes (for arbitrary exponents). Equivalently, any odd covering system---if one exists---must use moduli involving at least five distinct odd primes. The proof introduces a weight function method. Moduli are partitioned into prime-power towers and composites; the towers define a "weight region" W in Z/LZ via CRT, and a union bound shows the composites cover at most an R-fraction of W with R = 41/45 < 1 for the worst-case prime set {3,5,7,11}. This leaves at least L/40 integers provably uncovered. The same method yields R = 2/3 for three primes (a short self-contained proof) and extends to five primes via a three-level refinement---weight function, Bonferroni correction, and pigeonhole-forced collisions at prime 3---which proves the impossibility unconditionally for 98.2% of exponent configurations. The remaining 1.8% reduce to a CRT coverage maximality conjecture (NC ≤ 0), for which we provide an analytical proof at k ≤ 3 primes and exhaustive computational verification over 9,000,000+ exact configurations at k = 4 with zero violations.
Category: Number Theory
[3] viXra:2604.0023 [pdf] submitted on 2026-04-08 12:18:49
Authors: Hans Montanus
Comments: 8 Pages.
The successive action of the generators of the full modular group SL(2, Z) on thefundamental domain produces a tessellation of the upper half-plane H. Each tile is acurved triangle whose boundaries are circular arcs. We will analyze the curvatures ofthe boundaries from a flat space point of view. All Euclidean curvatures are integer.We will show that these integer curvatures are either odd or multiples of 8, and thatevery odd number and every multiple of 8 occurs as a curvature.
Category: Number Theory
[2] viXra:2604.0007 [pdf] submitted on 2026-04-03 23:58:54
Authors: Nikola Chachev
Comments: 15 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present a new algebraic reformulation of the uniqueness problem for periodic orbits of the Collatz map $c(n) = n/2$ ($n$ even) or $c(n) = 3n+1$ ($n$ odd). The question of whether ${1,2,4}$ is the only positive integer cycle is classically equivalent to an integer divisibility condition of the form $(2^S - 3^L) mid N$. We recast this condition as the vanishing of an explicit integer polynomial --- the cycle polynomial $P_G(t)$ --- evaluated at an arithmetic point $t_0$ of multiplicative order $L$ modulo $D = 2^S - 3^L$. This perspective reduces the uniqueness problem to a question of polynomial non-vanishing over $mathbb{Z}/Dmathbb{Z}$, which we analyse through the 2-adic and 3-adic structure of the evaluation map $G mapsto P_G(t_0) bmod D$. Using this framework we establish two partial results. First, for every mixed valuation sequence $G$ --- one in which the accumulated deviations $varepsilon_i$ take both positive and negative values --- the cycle polynomial satisfies $P_G(t_0) otequiv 0 pmod{D}$ in the special case where exact integer vanishing $A(G) = B(G)$ would be required; this follows from a parity obstruction on 2-adic valuations together with the step-size constraint $G_i geq 1$. Second, we identify a combined 2-adic and 3-adic obstruction that constrains any hypothetical solution $P_G(t_0) equiv 0 pmod{D}$ to an increasingly rigid arithmetic structure. The case of non-zero multiples --- whether $A(G) - B(G) = kD$ for $k geq 1$ --- remains open; we describe precisely thegap and the new ideas that would be needed to close it.
Category: Number Theory
[1] viXra:2604.0002 [pdf] submitted on 2026-04-02 21:37:03
Authors: Calvin Alexander Grant
Comments: 16 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
The Riemann Hypothesis is proved by showing that every non-trivial zero of the Riemannzeta function ζ(s) lies on the critical line Re(s) = 1/2. The proof is carried out entirely in the persistent-remainder category, where the relevant object is not a smooth local carrier but a graph-space with positive scaling excess. First, the prime-harmonic tail of ζ in the critical strip is shown to belong to the persistent-remainder class, so infinitesimal closure fails and tangent escape is blocked at all scales. Second, the graph-space obstruction is written in discrete form through Hessian sign phases, line-events, and closed sign chains: the line is not primitive, but is generated as the zero-interface between opposite sign phases, and the minimal half-turn-preserving chain forces the 1:3 topology. Third, the functional equation ξ(s) = ξ(1 − s) is identified as the arithmetic half-turn whose unique fixed locus is Re(s) = 1/2; by the half-turn sign law of the companion paper [1], any zero off this locus is a destructive node and is therefore inadmissible. Faltings’ theorem is retained as the arithmetic shadow of the same obstruction: in genus g ≥ 2, unrestricted rational refinement fails just as unrestricted infinitesimal smoothing fails in graph space. The same argument extends to Dirichlet L-functions, establishing theGeneralized Riemann Hypothesis.
Category: Number Theory
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