| viXra.org > Number Theory > 2411 |
 |
 |
| viXra.org > Number Theory > 2411 |
|
|
[22] viXra:2411.0174 [pdf] submitted on 2024-11-29 00:23:31
Authors: Edgar Valdebenito
Comments: 4 Pages.
In this note we give some formulas related to Prouhet-Thue-Morse constant.
Category: Number Theory
[21] viXra:2411.0167 [pdf] replaced on 2025-08-29 23:10:56
Authors: Mar Detic
Comments: 4 Pages.
This paper presents a detailed derivation and analysis of the seriesπ =∞Xk=0 12k + 1 + 24k + 1 + 14k + 3 − 14k.We demonstrate its equivalence to a definite integral, prove its convergence to π via analytical evaluation,and analyze its convergence rate, showing that it yields approximately 3 correct decimal digits per5 terms. Crucially, we show that the factor of (−1/4)k imbues the series with the property of a spigotalgorithm for the binary digits of π. While not as computationally efficient as the Bailey—Borwein—Plouffeformula, this series’ derivation from a simple rational function using undergraduate-level techniquesoffers significant pedagogical value in understanding the construction of digit-extracting algorithms forfundamental constants.
Category: Number Theory
[20] viXra:2411.0143 [pdf] submitted on 2024-11-22 21:39:45
Authors: Marko V. Jankovic
Comments: 6 Pages.
n this paper, a modification of Collatz method is going to be presented, It is going to be proved that iterative implementation of the corresponding modified Collatz function on any natural number will eventually lead to the number 1. An alternative definition of Collatz and modified Collatz functions are going to be presented, and some interesting results that are obtained from it are going to be briefly analyzed.
Category: Number Theory
[19] viXra:2411.0137 [pdf] submitted on 2024-11-21 21:58:20
Authors: Mostafa Senhaji
Comments: 10 Pages.
This article explores Dirichlet's theorem on arithmetic progressions and its connection with the twin prime conjecture. Dirichlet's theorem, proven in 1837, states that there are infinitely many prime numbers in each arithmetic progression of the form ��+����, when gcd(��,��)=1. This result is crucial for understanding the distribution of prime numbers in specific arithmetic sequences. The article delves deeper into the application of this theorem to the twin prime conjecture, which posits that there are infinitely many pairs of prime numbers whose difference is 2. Although this conjecture is still unsolved, it is based on principles similar to those used in the proof of Dirichlet's theorem. Using advanced mathematical tools, such as L(s, χ) functions and Dirichlet series, the paper proposes a new approach to formally demonstrate the existence of infinitely many pairs of twin primes. Furthermore, the article examines the generalization of this conjecture for any difference k, thus providing a broader perspective on the distribution of prime numbers in arithmetic progressions. By analyzing the non-nullity of L functions at ��=1, the article suggests the existence of infinite solutions to the twin prime conjecture and opens promising avenues to solve this problem as well as its generalizations.
Category: Number Theory
[18] viXra:2411.0128 [pdf] submitted on 2024-11-20 20:46:59
Authors: Philip Zixuan Pan
Comments: 5 Pages. (Note by viXra Admin: An abstract in the article is required)
Traditional research in number theory primarily revolves around the Eratosthenes sieve method and other methodologies, resulting in significant achievements. Yet certain challenging issues persist. For example, Goldbach’s Conjecture, has to be proven. This paper proposes an innovative approach and a formula for primes is established, supported with a Python-coded program.
Category: Number Theory
[17] viXra:2411.0119 [pdf] submitted on 2024-11-18 07:06:42
Authors: Lynette Michael Winslow
Comments: 15 Pages.
In this paper, we provide a detailed and rigorous proof that the series formed by the reciprocals of Sophie Germain primes and the series formed by the reciprocals of safe primes are both convergent. Utilizing analytic techniques and careful estimations of the counting functions for these primes, we establish upper bounds that demonstrate the convergence of these series.
Category: Number Theory
[16] viXra:2411.0113 [pdf] submitted on 2024-11-16 16:52:31
Authors: Roland Quême, Abdelmajid Ben Hadj Salem
Comments: 4 Pages. Submitted to Compositio Mathematica Journal. Comments welcome.
The aim of this article is to prove that all the roots of the Dirichlet eta function are only on the critical line, which implies that the Riemann hypothesis is true. The roots of the eta and zeta functions in the critical strip are the same. We shall investigate the possibility or not of existence of roots of the eta function which are in the critical strip and not on the critical line.
Category: Number Theory
[15] viXra:2411.0104 [pdf] replaced on 2024-11-15 19:25:11
Authors: Mohamed Chraiti
Comments: 10 Pages.
This article introduces a research program to study a new class of composite numbers, called Pseudo-Isolated Numbers (PIN), defined by their unique position between two consecutive prime numbers. where does it go to present its formal definition. This research program can open new perspectives in number theory and suggests potential applications in cryptography and algorithmic theory,...,etc,...
Category: Number Theory
[14] viXra:2411.0098 [pdf] submitted on 2024-11-14 20:40:45
Authors: Ramaswamy Krishnan
Comments: 5 Pages. (Note by viXra Admin: An abstract and scientific references in the article are required!)
First: it is proved for numbers for 1, 5, 21, 85..etc. Second: using the above, it is proved for n = 4k + 1. Third: n = 4k + 3 is analyzed for various values, and how it iterates first to 4k + 1 before becoming 1. Thus the conjecture is proved for all odd values of n. As even values of n are reduced to an odd value, the Collatz's Conjecture is proved for all positive integer values of n.
Category: Number Theory
[13] viXra:2411.0097 [pdf] replaced on 2024-11-17 08:16:42
Authors: Timothy Jones
Comments: 3 Pages. There was as error in the earlier draft that someone pointed out. This draft gives a correction.
It is shown that the limit of cos(j) and sin(j) as j goes to infinity do not exist. Using DeMoivre's theorem this implies the limits of sin(j!) and cos(j!) don't exist either. Assuming pi is rational, its multiple can be expressed as a factorial. From this a contradiction is derived.
Category: Number Theory
[12] viXra:2411.0079 [pdf] submitted on 2024-11-11 20:55:08
Authors: Budee U. Zaman
Comments: 11 Pages.
The Riemann Hypothesis remains one of the most critical unsolved problems in mathematics, proposing that all non-trivial zeros of the Riemann zeta function are located on the critical line. This paper explores unique fractal structures within the zeta function, drawing comparisons with the Mandelbrot set and the Smith chart, which together illuminate possible connections between prime distribution,electromagnetic symmetry, and gravitational principles.
Category: Number Theory
[11] viXra:2411.0071 [pdf] submitted on 2024-11-09 10:22:11
Authors: J. Kuzmanis
Comments: 45 Pages.
The rules describing emergence of abc-triples formed by the set of roots for the specificfamily of Pell’s equations x2 − Dy2 = ±4 are revealed.
Category: Number Theory
[10] viXra:2411.0068 [pdf] submitted on 2024-11-08 08:53:16
Authors: Hans Hermann Otto
Comments: 3 Pages.
This is a simple number-theoretical exercise for interested pupils presenting a connection between the golden mean power series expansion an real numbers.
Category: Number Theory
[9] viXra:2411.0053 [pdf] submitted on 2024-11-07 11:48:21
Authors: J. Kuzmanis
Comments: 54 Pages.
+/— choice sequences of consecutive odd primes, whose length is equal or exceeds 6,show strict regularities as integer representatives. This leads to conclusion that all even naturalnumbers above 118 can be written as sums of two different odd primes. Mentioned results arebased exclusively on Bertrand’s postulate.
Category: Number Theory
[8] viXra:2411.0052 [pdf] submitted on 2024-11-07 11:51:40
Authors: J. Kuzmanis
Comments: 28 Pages.
The rules describing emergence of abc-triples formed by the set of roots for Pell’sequations x2 − Dy2 = N with N = ±1 and N = ±2 are revealed.
Category: Number Theory
[7] viXra:2411.0040 [pdf] replaced on 2024-11-11 23:02:31
Authors: Jay Pillai
Comments: 7 Pages.
In this paper a theorem will be proven that is a relatively concise way to confirm the irrationality of an infinite series. From this, new proofs of the irrationality of some known results can be derived(pi, phi) as well as solutions to open problems(rationality of zeta function over the natural numbers, rationality of Euler-Mascheroni Constant)
Category: Number Theory
[6] viXra:2411.0036 [pdf] submitted on 2024-11-05 16:36:03
Authors: Dmitri Martila, Stefan Groote
Comments: 3 Pages.
A criterion given by Jean-Louis Nicolas is used to offer a proof for the Riemann Hypothesis in a straightforward way.
Category: Number Theory
[5] viXra:2411.0027 [pdf] submitted on 2024-11-04 18:02:14
Authors: Edgar Valdebenito
Comments: 2 Pages.
We give a sequence that converges to pi quickly.
Category: Number Theory
[4] viXra:2411.0017 [pdf] submitted on 2024-11-03 22:51:57
Authors: Mostafa Senhaji
Comments: 48 Pages.
The Riemann hypothesis is more than just a mathematical problem. It is an age-old enigma, a challenge to all those who dare to explore the depths of prime numbers. For more than 160 years, it has fascinated, intrigued, and resisted the attempts of the greatest minds. This document is the result of a passionate quest to elucidate this mystery. Here, each line traces a path through the complexities of the hypothesis, with the hope of shedding new light on this fundamental question. Whether you are an expert or simply curious to discover one of the greatest mysteries of the mathematical universe, I invite you to dive into this intellectual adventure. Together, let's explore the elegance hidden behind the formulas and perhaps get closer to the truth.
Category: Number Theory
[3] viXra:2411.0012 [pdf] replaced on 2025-07-19 22:35:47
Authors: Philippe Sainty
Comments: 56 Pages.
In this article the proof of the binary Goldbach conjecture is established (Any integer greater than one is the mean arithmetic of two positive primes) . To this end, Chen’s weak conjecture is proved (Any even integer greater than one is the difference of two positive primes) and a "localised" algorithm is developed for the construction of two recurrent sequences of primes ( U_2n ) and ( V_2n ), ( ( U_2n ) dependent of ( V_2n) ) such that for any integer n ≥ 2 their sum is equal to 2n : ( U_2n ) and ( V_2n ) are extreme Goldbach decomponents. To form them, a third sequence of primes ( W_2n) is defined for any integer n ≥ 3 by V_2n = Sup ( p ∈ P : p ≤ 2n - 3 ) , P denoting the set of positive primes. The Goldbach conjecture has been prove d for all even integers 2n between 4 and 4.1018. and in the neighbourhood of 10^100 , 10^200 and 10^300 for intervals of amplitude 10^9 . The table of extreme Goldbach decomposants, compiled using the programs in Appendix 14 and written with the Maxima and Maple scientific computing software, as well as files from ResearchGate, Internet Archive, and the OEIS, reaches values of the order of 2n = 10^5000. In addition, a global proof by strong recurrence "finite ascent and descent method" on all the Goldbach decomponents is provided by using sequences of primes (��2_q2n) defined by :��_q2n = Sup ( p ∈ P : p ≤ 2n - q ) for any odd positive prime q , and a majorization of U_2n by n^0.525 , 0.7ln_2.2(n) with probability one and 5 ln1.3(n) on average for any integer n large enough is justified.. Finally, the Lagrange-Lemoine-Levy (3L) conjecture and its generalization called "Bachet-Bézout-Goldbach"(BBG) conjecture are proven by the same type of method.
Category: Number Theory
[2] viXra:2411.0007 [pdf] replaced on 2025-11-14 00:14:07
Authors: Amine Oufaska
Comments: 6 Pages.
Since the ancient Greeks over 2000 years ago , mathematicians have asked the question about the law of distribution of the twin prime numbers . In this paper we present the set of definition and the Diophantine equation of the twin prime numbers , it is a magic subset of the natural numbers which arranges the twin prime numbers. This theory will provide a deep understanding of the twin prime conjecture and the Riemann hypothesis.
Category: Number Theory
[1] viXra:2411.0006 [pdf] submitted on 2024-11-01 20:39:07
Authors: Fian Qnoz
Comments: 6 Pages.
This short document purported to publicise assorted constants to be targeted for Reverse Arithmetic (known as unscrambling the egg in pop culture) experimentation. In section one, a tangential intermezzo intended to justify the urgency to archive and some trivia regarding Pie and its defense against the manifesto of otherwise obscure alternative. Later sections detail the constants aimed for Reverse Arithmetic experimentation.
Category: Number Theory
| Third-Party Links: |
|