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[22] viXra:2506.0143 [pdf] submitted on 2025-06-25 15:24:01
Authors: Theophilus Agama
Comments: 13 Pages.
We develop explicit bounds for the gap between consecutive terms in an addition chain leading to a fixed target 2^m<n<2^{m+1}.
Category: Number Theory
[21] viXra:2506.0124 [pdf] submitted on 2025-06-21 21:23:18
Authors: Giovanni Di Savino
Comments: 1 Page. (Note by viXra Admin: Please cite and list scientific references)
Fermat, studying Book II of Diophantus' Arithmetica, in the pages dedicated to the problems and observations around the Pythagorean Theorem, in a marginal note of the book, reports: "It is impossible to write a cube as the sum of two cubes or a fourth power as the sum of two fourth powers or, in general, no number that is a power greater than two can be written as the sum of two powers of the same value". This conjecture, known for many years as Fermat's last theorem, became a theorem because Prof. Andrew Wiles demonstrated that c^n≥3 ≠ a^n≥3 + b^n≥3
Category: Number Theory
[20] viXra:2506.0112 [pdf] submitted on 2025-06-20 20:12:44
Authors: Theophilus Agama
Comments: 6 Pages.
Let us define the function F(m, r) as the number of integers n in the interval [2^m, 2^{m+1}) such that ι(n) ≤ floor(m + r), where r = c·m / log m for some constant c satisfying 0 larger than c smaller than log 2. Next, define α = c + log 2 − (1/4)(1 − o(1)). By applying the Chernoff inequality from probability theory and using ideas from De et al. (2025), we obtain the following improved upper bound: F(m, c·m / log m) ≤ exp[ α·m − (1 − ε)·(c·m·log log m) / log m ] for any small ε > 0 as m tends to infinity.
Category: Number Theory
[19] viXra:2506.0110 [pdf] submitted on 2025-06-20 20:10:04
Authors: Theophilus Agama
Comments: 6 Pages.
Define the function F(m, β(m) − m) as the number of integers n in the interval [2^m, 2^{m+1}) such that l(n) ≤ β(m). By applying ideas from De Koninck et al. (2025), we obtain the following general upper bound: F(m, β(m) − m) ≤ exp[ (β(m) − m) · (2·log β(m) + (1 + ε)·2·log log m + O(1)) ] for any small ε > 0 as m tends to infinity. This result generalizes a recent bound proved in the work of De Koninck and collaborators.
Category: Number Theory
[18] viXra:2506.0109 [pdf] submitted on 2025-06-19 20:58:23
Authors: Yibing Xiong
Comments: 84 Pages. (Note by viXra Admin: Please cite and list scientific references)
Disordered events can be divided into two types: disordered deterministic events - generalized events, disordered random events - random events - narrow events; Establishing probability axioms, boundary axioms, etc., proposing four important original works: probability method, boundary, subdivision probability, Xiong's sieve; Establish General Probability Theory. Using analytical methods to solve mathematical problems such as the prime number theorem, Riemann hypothesis, twin prime numbers, densest K-tuple prime numbers, K-tuple prime numbers, Goldbach's conjecture, Gaussian lattice points, etc.
Category: Number Theory
[17] viXra:2506.0096 [pdf] submitted on 2025-06-18 19:43:06
Authors: Theophilus Agama
Comments: 9 Pages. (Note by viXra Admin: Please don't use LaTeX codes in the abstract!)
[This note provides a quantitative bound for the number of primes in an addition chain]
Category: Number Theory
[16] viXra:2506.0080 [pdf] submitted on 2025-06-16 01:24:22
Authors: Ahmed Souissi
Comments: 3 Pages.
Abstract. While exploring Dirichlet L-functions as part of a final-year project, I stumbledupon a surprising property: a wave function ψp(x) = χ(p)eiγx, where χis a non-trivial Dirichlet character modulo q, γ is a non-trivial zero of L(s,χ), and p is a prime, produces a discrete Fourier transform ψp(k) with a dominant peak at k ≡ p−1 mod q. This "mirror symmetry"suggests a deep arithmetic structure linking primes to their modular inverses. I formalize this observation with a quantitative conjecture, provide numerical evidence for q = 5,13,17, andoffer a partial theoretical analysis using Gauss and Kloosterman sums. Potential applicationsin quantum physics and cryptography are discussed.
Category: Number Theory
[15] viXra:2506.0079 [pdf] replaced on 2025-06-23 00:42:25
Authors: Walter A. Kehowski
Comments: 8 Pages. The abstract was rewritten, a proof of the Euclid-Euler Theorem was included, and some typos corrected. Comments welcome.
A sublime number is a number such that both its number of divisors and sum of divisors are both perfect numbers. The number 12 is the first sublime number. This paper gives Kevin Brown's construction of even sublime numbers a modern mathematical development.
Category: Number Theory
[14] viXra:2506.0055 [pdf] submitted on 2025-06-12 22:24:39
Authors: Taha M. Muhammad
Comments: 4 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
a, b, c, d, e, f, g ∈ N_+ ⟺ Euler Perfect Box Let a, b, c, d, e, f, r, k ∈N_+. I have to prove that g ∈N_+ Let a < b < c, & [(a
Category: Number Theory
[13] viXra:2506.0054 [pdf] submitted on 2025-06-12 22:24:29
Authors: Taha M. Muhammad
Comments: 10 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
1- Let x ∈ Loop of Collatz Sequence = LS(n), n, x,y,z,t,u,v ∈ N+ i) I proved LS(n) = { }, {x}, {x,y}, {x,y,z,t}, {x,y,z,t,u,u2026,v }, and {x,y,z,t,u,u2026,v,u2026} is false. ii) I proved LS(n) = {x,y,z} = {4,2,1} = {4,2,1, 4,2,1} =u2026= {4,2,1, u2026, 4,2,1} is true ∀n ∈ N+ 2- I created special sketch to find algebra expressions equal to x. 3- I created special Table to find values of elements in the LS(n). 4- I created a pattern to find values of x ∈ LS(n). 5- Observation: let r, k ∈ N+, r = number of elements in the LS(n):i) if r = 3k ⇒ x ∈ N+ ⇒ LS(n) = {1,2,4} ∀n ∈ N+ ii) if r = 3k + h, such that h=1,or 2⇒ x ∉ N+ ⇒ LS(n) has no loop ∀n ∈ N+ ∴ LS(n) ={4,2,1},∀n ∈ N+.
Category: Number Theory
[12] viXra:2506.0053 [pdf] submitted on 2025-06-12 22:36:30
Authors: Taha M. Muhammad
Comments: 5 Pages. (Note by ai.viXra.org Admin: Author name should be listed after the title; please cite and list sceintific references)
Fermat’s Last Theorem: There are no natural numbers (1, 2, 3, u2026) a, b, and c such that a^n+b^n = c^n, in which n is a natural number greater than 2. 1-Taha^' s Logic Equations Fact a=b (True or False) & c=d (True)⇒(a-c=b-d) Fulse⇔a=b ( False) 2- Taha^' s Coefficient Fact: [a + b = c ⇒ ax + by= cz] ⇔ x = y = z3- Taha’s (N_+) & Three-Sided Geometric Shapes Fact below: N_+ = (Right Triangles) ∪(Acute Triangles) ∪(Obtuse Triangles) ∪ (Segments c=a + b) ∪ (Segments c>a + b).
Category: Number Theory
[11] viXra:2506.0052 [pdf] replaced on 2025-12-14 05:44:37
Authors: Hajime Mashima
Comments: 82 Pages.
General solution conditions applies when the equation of Fermat's proposition can be phase-transformed by a periodic product.
Category: Number Theory
[10] viXra:2506.0042 [pdf] submitted on 2025-06-10 22:32:14
Authors: Edgar Valdebenito
Comments: 3 Pages. (Note by viXra Admin: Further repetition may not be accepted)
We give some formulas related to the number 239.
Category: Number Theory
[9] viXra:2506.0041 [pdf] submitted on 2025-06-09 20:49:25
Authors: Ahcene Ait Saadi
Comments: 7 Pages. (Note by viXra Admin: Please cite and list relevant scientific references)
In this article, I studied the behaviour of natural integers, after performing operations on them. The results I have obtained are intriguing and I cannot explain them. I share this original work with young researchers for deepening and improvement.Key words: Integers, operations, sequences.
Category: Number Theory
[8] viXra:2506.0030 [pdf] submitted on 2025-06-08 19:30:38
Authors: Jean-Yves Boulay
Comments: 40 Pages.
Here is expanded the concept of Sophie Germain prime and safe prime to ultimate numbers. More, this mathematical mechanism is also applied to the set of all whole numbers (ℕ) which are differentiated into ultimate and non-ultimate numbers. Thus, a complete study of the set ℕ is undertaken. This global investigation, universally broadening the mechanical-mathematical concept of Sophie Germain, makes it possible to propose a genetics of numbers very similar to biological genetics. According to these new numeric genetic criteria, in their start top organization, geometric distribution of whole numbers in various closed matrices, is organized into perfect ratios to exact 3/2 or 1/1 value.
Category: Number Theory
[7] viXra:2506.0024 [pdf] submitted on 2025-06-07 20:13:06
Authors: Abdelrahman M. Mohammed
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We begin by observing a striking "mirror-complement" pattern in the binary digits of √2: whenever, at any position, a run of k equal bits is separated by a single opposite bit from another run of bits, those two runs must have equal length. Restricting to prime-indexed positions, the same pattern remains perfectly true for millions of primes. This phenomenon is a direct consequence of the classical digit-by-digit square-root algorithm in base 2, because each comparison uses4Pn + 1 = 2 (2Pn) + 1,i.e. "copy + complement + copy."From this insight we build a two-rectangle coding on T² whose itinerary reproduces the binary digits of √2. A measurable conjugacy to the (½,½) Bernoulli shift allows us to apply Chung—Smorodinsky’s bounded-coboundary theorem (1967), showing each cylinder-indicator has a uniform sup-norm bound. Telescoping that coboundary yields a universal O(Nu207b¹) discrepancy bound on every length-ℓ binary block, proving base-2 normality of √2. Finally, van der Corput differencing and Wall’s criterion transfer the same O(Nu207b¹) bound to every integer base B ≥ 2, establishing that √2 is normal in all bases.This paper unifies these ideas—starting from the prime-indexed mirror pattern and culminating in a gap-free, self-contained proof of full normality of √2.
Category: Number Theory
[6] viXra:2506.0023 [pdf] submitted on 2025-06-07 20:07:13
Authors: Viktor Arvidsson
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We prove the twin prime conjecture, asserting infinitely many integers x such that both x and x + 2 are prime. Construct an N × N diagonal matrix HN with diagonal entries Dx = 0 if x, x + 2 prime, and Dx ≥ 3 10 otherwise. The kernel dimension equals the number of twin primes up to N, denoted |TN|. A heat-trace argument on Tr[exp(−2HN)] forces |TN| → ∞, proving the theorem. We refine parameter bounds, standardize notation, and add explicit estimates for key constants.
Category: Number Theory
[5] viXra:2506.0019 [pdf] replaced on 2025-06-15 01:54:23
Authors: Hans Hermann Otto
Comments: 9 Pages.
We show the importance of Lucas number 123, symbolized by others for divine order or trinity, as enigmatic number for life, physics and the cosmos. This number is related to fundamental constants of nature and to powers of the golden mean that governs phase transitions from particle to cosmic scale. It is also related to the Higgs boson and to the Great Pyramid. Our number theoretical approach may help to understand unsolved problems in physics and supports applications.
Category: Number Theory
[4] viXra:2506.0018 [pdf] submitted on 2025-06-06 19:47:59
Authors: Fabrice Trifaro
Comments: 45 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Using a comprehensive approach, this paper aims to demonstrate, clearly and rigorously, the validity of the Collatz conjecture. To this end, the original 3n+1 iteration is reformulated by isolating the odd terms into sequences referred to as R-Cz sequences. These sequences are analyzed through their structural properties and their distribution among the odd natural numbers. As a first essential result, it is shown that they do not admit non-trivial cycles: the only possible cycle is the trivial one, of value and length 1. Two independent proofs that all R-Cz sequences converge are then presented. The first, combinatorial in nature, relies on the finiteness of intervals that could possibly separate terms of the sequences. The second, set-theoretic, is based on a contradiction between the countability of the odd integers and the uncountable cardinality of the hypothetical divergent R-Cz sequences. Both methods lead to the same conclusion: all Collatz sequences eventually enter the cycle (1,4,2).
Category: Number Theory
[3] viXra:2506.0009 [pdf] submitted on 2025-06-03 23:39:40
Authors: Predrag Terzić
Comments: 2 Pages. (Note by viXra.org Admin: Please cite and list sceintific references)
We present new continued fraction representation of constant pi.
Category: Number Theory
[2] viXra:2506.0006 [pdf] submitted on 2025-06-02 19:37:53
Authors: Predrag Terzić
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
We present a new, specific primality test for numbers of the form N = 4p^n - 1, where p is an odd prime and n > 0. The test is a generalization of the Lucas-Lehmer test for Mersenne numbers and relies on a sequence defined by Dickson polynomials. We prove that, under a certain condition, N is prime if and only if the n-th term of a specific sequence is congruent to zero modulo N. This provides a deterministic primality test for this family of numbers.
Category: Number Theory
[1] viXra:2506.0001 [pdf] replaced on 2025-07-23 20:25:30
Authors: Mar Detic
Comments: 3 Pages.
We introduce a primality testing framework based on examining the greatest common divisors (GCDs) of a candidate integer p with terms froma quadratic sequence defined by Sq = q2 + 2q for odd integers q. This approach generalizes and extends classical primality criteria by leveraging properties of quadratic forms. We formalize this method as a conjecture,analyze its computational complexity, discuss potential error cases such as pseudoprimes, and present empirical validations demonstrating its effectiveness.
Category: Number Theory
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