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[24] viXra:2206.0168 [pdf] replaced on 2022-07-18 07:37:56
Authors: Marco Rolando Burgos Chambi
Comments: 13 Pages.
On 1859, the german mathematician Georg Friedrich Bernhard Riemann made one of his most famouspublications "On the Number of Prime Numbers less than a Given Quantity" when he was developing hisexplicit formula to give an exact number of primes less than a given number x, in which he conjectured that"all non-trivial zeros of the zeta function have a real part equal to 1/2 ". Riemann was sure of his statement,but he could not prove it, remaining as one of the most important hypotheses unproven for 163 years.In this paper, we have to prove that the Riemann Hypothesis is true, based on the Bernoullipower sum, the Euleru2013Maclaurin formula and its relation with the Riemann Zeta function.
Category: Number Theory
[23] viXra:2206.0164 [pdf] replaced on 2022-07-26 12:38:41
Authors: Timothy W. Jones
Comments: 4 Pages. Thanks for any patience. I think this draft should do it.
A direct proof shows Goldbach's conjecture is correct. It is as simple as can be imagined. A table consisting of two rows is used. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. Without loss of generality, an example shows that primes, ones that sum to 2n will always be left in such columns.
Category: Number Theory
[22] viXra:2206.0158 [pdf] submitted on 2022-06-29 07:50:37
Authors: Daoudi Rédoane
Comments: 1 Page.
Here I present an interesting equality between a continued fraction where the arctan function is involved and pi.
Category: Number Theory
[21] viXra:2206.0140 [pdf] submitted on 2022-06-26 17:54:54
Authors: Daoudi Rédoane
Comments: 3 Pages.
Here I present one formula that produces prime numbers.
Category: Number Theory
[20] viXra:2206.0139 [pdf] submitted on 2022-06-26 18:10:40
Authors: Daoudi Rédoane
Comments: 3 Pages.
Here I present a conjecture about Euler's totient function and primes.
Category: Number Theory
[19] viXra:2206.0130 [pdf] submitted on 2022-06-24 07:37:00
Authors: Marko V. Jankovic
Comments: 25 Pages.
In this paper a proof of the Polignac's Conjecture is going to be presented. The proof represents an extension of the proof of the twin prime conjecture. It will be shown that primes with a gap of size g could be obtained through
two stage sieve process, and that will be used to prove the conjecture.
Category: Number Theory
[18] viXra:2206.0129 [pdf] submitted on 2022-06-24 10:01:51
Authors: Roberto Compare
Comments: 2 Pages.
Choose a positive integer,
if it is even divide it by 2, if it is odd multiply it by 3 and add 1, repeat. What is the final result? This problem is known as the 3n+1 problem and the Collatz Conjecture states that no matter the initial value you will always reach the cycle 1,4,2.
Category: Number Theory
[17] viXra:2206.0114 [pdf] replaced on 2022-06-23 09:34:04
Authors: Timothy W. Jones
Comments: 10 Pages. A few minor corrections and tweaks.
We show all all reduced c/d in (0,1) can be represented using any base any positive integer b greater than 1.
Category: Number Theory
[16] viXra:2206.0097 [pdf] submitted on 2022-06-18 09:41:47
Authors: Jesus Sanchez
Comments: 26 Pages.
In this paper we will demonstrate that we can check if a number is prime, the number of factors it has and even information about these factors using the following integral:
q=1/2π ∫_(-π)^π〖e^pjω (L_(〖(e〗^(-jω))) (s,2)-e^(-2jω)/(1-e^(-jω) )) 〗 dω (1)
Where p is the number to be checked and L_(〖(e〗^(-jω))) (s,2) represents the Lambert series that will be studied in the paper. All the steps to arrive to that integral will be shown although they will be very similar to the already shown in [12][13].
In this paper to calculate the Lambert series apart from the summation that comes from its definition, it will be used the paper by Banerjee-Wilkerson [14] that provides closed solutions for different cases.
In the conclusions, it will be shown if this exercise is computationally worth to make the primality test and calculate the factors of a number p.
Category: Number Theory
[15] viXra:2206.0094 [pdf] submitted on 2022-06-18 14:06:34
Authors: Bertrand Wong
Comments: 36 Pages.
This informative paper, which is published in an international mathematics journal, presents insights and many important points on the prime numbers, which are the building-blocks or “atoms” of the integers, and the Goldbach conjecture formulated by Christian Goldbach (1690 - 1764) which are the result of years of research (the author having published two papers on the Goldbach conjecture in an international mathematics journal in 2012), all of which would be of interest to researchers working on the prime numbers and the Goldbach conjecture itself. The Goldbach conjecture, viz., every even number after 2 is the sum of 2 primes, is actually related to the distribution or “behavior” of the prime numbers. Therefore, when the distribution or “behavior” of the prime numbers is firmly understood the conjecture could be more easily solved. The paper has much to share about the distribution or “behavior” of the prime numbers, providing much numerical evidence to support the conjecture, besides suggesting ways or arguments for resolving the conjecture.
Category: Number Theory
[14] viXra:2206.0093 [pdf] submitted on 2022-06-18 14:11:37
Authors: Bertrand Wong
Comments: 16 Pages.
This paper, which is published in an international mathematics journal, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.
Category: Number Theory
[13] viXra:2206.0090 [pdf] submitted on 2022-06-17 20:46:38
Authors: Palmioli Luca
Comments: 7 Pages. (Corrections made by viXra Admin to conform with the requirements on the Submission Form)
The purpose of this study was to rewrite the formulas for the sum of powers of integers in a subsequent general mathematical formula independent of Bernoulli polynomials and numbers, starting from the formula of Faulhaber.
Category: Number Theory
[12] viXra:2206.0089 [pdf] replaced on 2022-06-20 04:46:20
Authors: Palmioli Luca
Comments: 8 Pages.
This study aims to bring to the knowledge of the scientific mathematical community of mathematical formulas that calculate the integer and fractional part of a positive or negative function.
Category: Number Theory
[11] viXra:2206.0084 [pdf] submitted on 2022-06-17 10:33:05
Authors: Giovanni Di Savino
Comments: 13 Pages. Goldbach and Euclid's twin primes measured on the plane
Abstract: Thales (1), by measuring the height of the inaccessible pyramid and the distance of the unreachable ship far from the port, demonstrates (1a) that anything that can be reported on a plane can be measured; Euclid (2) with the Theorem of the infinity of prime numbers proved that, however large the measure, a number n, there always exists a prime number greater than n and it has been proved that prime numbers are infinite; Gauss (3) with the Fundamental Theorem of Arithmetic (4) (hereinafter T.F.A.) proved that every natural number greater than 1 is either a prime number or can be expressed as a product of prime numbers. Goldbach's conjecture (5) states that the infinite natural numbers are the sum of only two or three numbers among the infinite primes and can be measured on the plane with the triangles of Viviani's Theorem (6) where the sum of two points of the xn + yn plane, shown on the abscissa and ordinate of the plane, are the distances of the infinite natural numbers from the sides of the triangle of Viviani's theorem (6a).
Category: Number Theory
[10] viXra:2206.0081 [pdf] submitted on 2022-06-15 00:36:28
Authors: Tsuneaki Takahashi
Comments: 4 Pages.
Number series of Collatz conjecture reaches to value 1 finally if the series of number has no looping. This has been mentioned statistically on viXra:2204.0151 (*1).
There is no looping in Number series of Collatz conjecture. This has been mentioned algebraically on viXra:2206.0056 (*2).
Here will try to prove algebraically Collatz conjecture is correct.
Category: Number Theory
[9] viXra:2206.0075 [pdf] submitted on 2022-06-15 20:39:42
Authors: Ronald Danilo Chávez Calderón
Comments: 38 Pages.
This paper shows a very elementary way of counting the number of primes under a given number with total accuracy. Is the function π(x) if 25 ≤ x ≤ 1572.
Category: Number Theory
[8] viXra:2206.0068 [pdf] submitted on 2022-06-13 20:56:21
Authors: Remy Aumeunier
Comments: 5 Pages.
In this paper, I propose a method to decompose all the integer in sum of 2 prime number, then justify the method.
Category: Number Theory
[7] viXra:2206.0051 [pdf] replaced on 2023-12-21 01:48:32
Authors: Toshihiko Ishiwata
Comments: 26 Pages.
This paper is a trial to prove Riemann hypothesis accordingto the following process.1. We make one identity regarding x from one equation that gives Riemannzeta function ζ(s) analytic continuation and 2 formulas (1/2+a±bi, 1/2−a ± bi) that show non-trivial zero point of ζ(s).2. We find that the above identity holds only at a = 0.3. Therefore non-trivial zero points of ζ(s) must be 1/2 ± bi because a cannothave any value but zero.
Category: Number Theory
[6] viXra:2206.0047 [pdf] submitted on 2022-06-10 20:48:37
Authors: Ryujin Choe
Comments: 1 Page. (Corrections made by viXra Admin to conform with the requirements on the Submission Form)
In this paper, we show that Robin inequation is true for large N, and RH is true also.
Category: Number Theory
[5] viXra:2206.0046 [pdf] submitted on 2022-06-10 15:26:13
Authors: Edgar Valdebenito
Comments: 5 Pages.
Some formulas related to Pi.
Category: Number Theory
[4] viXra:2206.0035 [pdf] replaced on 2023-02-24 06:23:30
Authors: Wing K. Yu
Comments: 11 Pages.
In this paper, we are going to prove Legendre’s Conjecture: There is a prime number between n^2 and (n+1)^2 for every positive integer n. We will also prove three related conjectures. The method that we use is to analyze a binomial coefficient. It has been developed from the method of analyzing a central binomial coefficient that was used by Paul Erdős to prove Bertrand’s postulate - Chebyshev’s theorem.
Category: Number Theory
[3] viXra:2206.0030 [pdf] submitted on 2022-06-06 19:49:11
Authors: Federico Romagnoli
Comments: 21 Pages. In English and Italian
It was decided to title this paper "The chaotic house of primes and the unprovable Riemann hypothesis" since the aim is to photograph the structure in which the prime numbers are placed (the house), to analyse the results obtained in the context of the Riemann hypothesis, results that attest to its non-verifiability, and finally to put emphasis on the distribution of primes, a distribution that is not random, not regular, but chaotic and from which order is generated.
Taking advantage the complementarity of the set of prime numbers with that of composite numbers, it is possible to photograph the structure in which primes are placed using the ordered structure of composite numbers. The latter is described by two families of double sequences (x,y) defined in Z_(x>l; y>m)^2→N_(>8) and whose analytical expressions change according to the choice of parameters l,m∈Z. Graphical representations follow to better grasp the regularity of composite numbers, as well as the spaces left by them free where the prime numbers find "house".
It is therefore possible to shed light on the distribution of primes and at the same time validate and contradict a particular aspect of the Riemann, namely that prime numbers are distributed with regularity. It follows the thesis according to which the Riemann hypothesis is impossible to prove or to disprove due to the "falsely ordered" nature of prime numbers. In fact, their nature is neither ordered nor random, but chaotic and generative of the order, the true one of composite numbers which, due to their complementarity with prime numbers, make appear ordered even the prime numbers. These are the conclusions.
Category: Number Theory
[2] viXra:2206.0018 [pdf] replaced on 2022-06-07 11:54:06
Authors: Lucian M Ionescu
Comments: 8 Pages.
A line of study of the Riemann Hypothesis is proposed, based on a comparison with Weil zeros and a categorification of the duality between Riemann zeros and prime numbers.
The three case of coefficients, complex, p-adic and finite fields are also related.
Category: Number Theory
[1] viXra:2206.0006 [pdf] submitted on 2022-06-01 20:02:43
Authors: Juan Elias Millas Vera
Comments: 5 Pages.
In this paper I want to expose the possibility of making a formula which counts the exact number of primes less than a given number, using some tools in analysis of functions, some tools in series of functions and some new tools.
Category: Number Theory
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