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[23] viXra:2209.0167 [pdf] submitted on 2022-09-29 11:59:06
Authors: Suaib Lateef
Comments: 4 Pages.
We pose a problem which is motivated by Newton's identity on sums of powers. We prove two special cases using algebraic manipulations. The method used is inefficient to prove all cases.
Category: Number Theory
[22] viXra:2209.0148 [pdf] replaced on 2022-10-24 13:49:33
Authors: Ahmed Idrissi Bouyahyaoui
Comments: 3 Pages. correction and improvement
Let x^n = z^n - y^n and x^(n-1) = az^(n-1) - by^(n-1), (a, b) Є Z^2.In the division with zero remainder of ab(z^n - y^n) by az^(n-1) - by^(n-1), it exists one and only one remainder equal to zero and valid, and then implies the equality b^2y^(n-2) = a^2z^(n-2) which is impossible for n > 2 since x^(n-1) = az^(n-1) - by^(n-1) and x, y, z are coprim integers.
Category: Number Theory
[21] viXra:2209.0147 [pdf] replaced on 2022-10-26 19:14:36
Authors: Ahmed Idrissi Bouyahyaoui
Comments: 3 Pages. correction and improvement / correction et amélioration
Let x^n = z^n - y^n and x^(n-1) = az^(n-1) - by^(n-1), (a, b) Є Z^2.In the division with zero remainder of ab(z^n - y^n) by az^(n-1) - by^(n-1), it exists one and only one remainder equal to zero and valid, and then implies the equality b^2y^(n-2) = a^2z^(n-2) which is impossible for n > 2 since x^(n-1) = az^(n-1) - by^(n-1) and x, y, z are coprim integers.
Category: Number Theory
[20] viXra:2209.0133 [pdf] replaced on 2023-12-08 16:04:42
Authors: Marco Burgos
Comments: 16 Pages.
One of the most famous functions, full of enigmas, is the Riemann zeta function, since it is the basis for one of the most amazing hypotheses because of its relation with the prime counting function. But in the present paper, we will discover some of those enigmas, using the Euler-Maclaurin formula.
Category: Number Theory
[19] viXra:2209.0124 [pdf] submitted on 2022-09-22 16:36:44
Authors: Edgar Valdebenito
Comments: 3 Pages.
In this note we give some formulas related to Catalan constant.
Category: Number Theory
[18] viXra:2209.0107 [pdf] replaced on 2023-11-14 22:12:37
Authors: Chen-Li Pan
Comments: We add Lemma 2.5 at page 12 to explain the formulation of (2.22) in Section 2, rewrite the 2nd paragraph of page 41 and refer it to the convention with the number l=1 for which it is built up at 3rd and 4th paragraphs of page 38.
This paper offers a breakthrough in proving the veracity of original Riemann hypothesis, and extends the validity of its method to include the cases of the Dedekind zeta functions, the Hecke L-functions hence the Artin L-functions, and the Selberg class.First we parametrize the Riemann surface $mathbf{S}$ of $log$-function, with which we first shrink the scale of each chosen parameter for which it depends on the chosen natural number $Q_{N_{0}}$ which is a chosen common multiple of all the denominators which are derived from a pre-set choice of rational numbers which approximate the values $log(k+1)$ with the integers $k$ in $0leq kleq N$.Then in (1.7) we define the mapping $-Q_{N_{0}}log(.)$ to pull the truncated Dirichlet $eta$-function $f_{N}(s)$ back to be re-defined on $mathbf{S}$, after that we shrink all the points to have their absolute values are all less than $1$ and closer to $1$. We apply the Euler transformation to the alternative series of Dirichlet $eta$-functions $f(s)$ which are defined in (1.4), then we build up the locally uniform approximation of Theorem 4.7 for $f(s)$ which are established on any given compact subset contained in the right half complex plane.In the second part we define the functions $phi(s)$ which are formulated in (6.1) then by specific property of the functions $phi(s)$, we have the similar asymptotics Theorem 6.5 as those of Theorem 4.7 to obtain the result of Theorem 6.8.And with the locally uniform estimation Lemma 5.10, finally in Theorem 5.9 and Theorem 6.9 we employ Theorem 5.8 and Theorem 6.8 to solve problems of Riemann hypothesis for the Dedekind zeta functions, the Hecke L-functions, the Artin L-functions, and the Selberg class for which all of their nontrivial zeros are contained in the vertical line $Re(s)=1/2$.Finally for the $gamma(s)$-factor of each Dirichlet series $D(s)$ which is formulated in (1.4), then by Theorem 6.9 it has neither zeros nor poles contained in the critical strip $0<Re(s)<1$ and the non-existence of Siegel's zeros for such Dirichlet series $D(s)$ is confirmed.
Category: Number Theory
[17] viXra:2209.0104 [pdf] submitted on 2022-09-18 01:31:37
Authors: Fabien Sabinet
Comments: 2 Pages.
A classic example of the Riemann series theorem use the alternating harmonic series that converges to lnu2061(2) when k → ∞ and when rearranged converges now to a different value 1/2 lnu2061(2) when k → ∞. But here I demonstrate that the rearranged series always exclude terms that constitute a third series which itself converges to exactly the difference between the original series and the rearranged one and thus explains why the rearranged series do not converge toward the original value. Eventually, it demonstrates also that the theorem is certainly false.
Category: Number Theory
[16] viXra:2209.0101 [pdf] replaced on 2022-10-12 18:19:32
Authors: Mohammed Bouras
Comments: 7 Pages. The sequence contain only ones and the primes in order (conjecture 3.1 and 3.2).
In this paper, we present a new sequence containing only ones and the prime numbers, which can be calculated in two different ways, the first way using the greatest common divisor (gcd) and Kurepa left factorial function, the second way consisting of using the denominator of the continued fraction
Category: Number Theory
[15] viXra:2209.0099 [pdf] submitted on 2022-09-17 00:36:47
Authors: Julian Beauchamp
Comments: 2 Pages. (Note by viXra Admin: Author name is required on the Article - Please conform!)
In this short paper, I prove that the difference between the square roots of two consecutive triangular numbers tends to √2/2, the reciprocal of √2, as the triangular numbers tend to infinity. I believe this relationship between √2 and triangular numbers is previously unknown.
Category: Number Theory
[14] viXra:2209.0074 [pdf] submitted on 2022-09-12 10:48:04
Authors: Seung-pyo Hong
Comments: 4 Pages.
In order to judge the prime number, many division operation is done. Those who read this document will be able to subtract by addition and subtraction without multiplication and division operations. A prime number is one with no proper factors. It would take a few hundred digits to determine which number is a prime number, and it would take a while to divide it into computer operations. It is almost impossible to determine whether tens of millions of digits are a minority. If the number of judgments is increased by dividing the decimal judgments, the processing time is increased sharply and the time can not be processed. However, the time of judgment is reduced by treating the small factor decomposition in another way.
Category: Number Theory
[13] viXra:2209.0070 [pdf] submitted on 2022-09-11 02:51:04
Authors: Zhi Li, Hua Li
Comments: 3 Pages.
The Legendre conjecture was proposed by the French mathematician Legendre (1752-1833) andhas not been proved for nearly 200 years. The conjecture is that between any two adjacent perfect square numbers, there is at least one prime number. That is, for any positive integer n, there is a prime number p such that n^2 < p < (n+1)^2.For the distribution of prime number is a distribution of deterministic random, problems related to prime numbers can be studied, analyzed and proved by probability statistics.This paper proves the conjecture by the method of probability and statistics, and proves that the number of prime numbers in the interval from n^2 to (n+1)^2 is similar to the number of prime numbers smaller than the integer n.
Category: Number Theory
[12] viXra:2209.0061 [pdf] replaced on 2024-10-13 14:19:45
Authors: V. Barbera
Comments: 14 Pages.
This paper presents a refinement of the Sieve method of Eratosthenes in conjunction with wheel factorization. The result is to use less memory.
Category: Number Theory
[11] viXra:2209.0059 [pdf] submitted on 2022-09-09 15:59:36
Authors: Zhi Li, Hua Li
Comments: 4 Pages.
The n^2+1 conjecture states that there are infinitely many natural numbers n such thatn^2+1 is a prime number.This paper defines the distribution type of prime numbers as deterministic randomdistribution. The distribution is characterized by limited degrees of freedom and a certain degree of predictability. This paper proves that the number of prime numbersin an interval is equivalent to the cumulative probability value. According to this, thenumber of prime numbers in a certain region can be determined by calculating thecumulative probability value. Therefore, the problems related to prime numbers can be studied, analyzed and proved by using probability and statistics methods.The conjecture is proved by judging the convergence of the series using probabilitystatistics method.
Category: Number Theory
[10] viXra:2209.0058 [pdf] submitted on 2022-09-09 17:29:22
Authors: Zhi Li, Hua Li
Comments: 2 Pages.
The natural numbers defined by the sequence F(n) =2^(2^n) + 1, n = 0,1,2,..., are calledFermat numbers. Fermat's conjecture states that there are only finitely many primenumbers in Fermat numbers. It was proposed in 1640 and has not been proved for more than 380 years.The prime number distribution is a deterministic random distribution, so problemsrelated to prime numbers can be studied, analyzed and proved by probability statistics.In this paper, the probability and statistics method is used to prove the conjecture byjudging whether the series converges. Our new conjecture is that there are only five known Fermat primes, namely 3, 5, 17, 257, and 65537.
Category: Number Theory
[9] viXra:2209.0053 [pdf] submitted on 2022-09-08 16:26:52
Authors: Stefano Barretta
Comments: 2 Pages.
This article shows that the series of odd square numbers is equal to the progression of the numbers of 8.
Category: Number Theory
[8] viXra:2209.0051 [pdf] submitted on 2022-09-08 21:08:28
Authors: Kouider Mohammed Ridha
Comments: 2 Pages.
In this paper, we interesting in most conjecture problem relies with the prime number which is Legendre's conjecture. We also introduced polynomials that check this conjecture with algebraic proof. Also, we reinforced the conjecture with some rules.
Category: Number Theory
[7] viXra:2209.0030 [pdf] submitted on 2022-09-05 19:21:41
Authors: Theophilus Agama
Comments: 5 Pages.
We apply the notion of the textbf{olloid} to show that a certain set contains no solution of the ErdH{o}s-Straus equation.
Category: Number Theory
[6] viXra:2209.0018 [pdf] submitted on 2022-09-03 23:25:56
Authors: Zhi Li, Hua Li
Comments: 4 Pages.
The Mersenne Prime Conjecture was proposed in 1644, which refers to whether there areinfinitely many Mersenne primes in a positive integer of the form 2^n- 1. The distribution of prime numbers is a deterministic random distribution, so problems related to prime numbers can be studied, analyzed and proved by probability statistics. This paper proves the conjecture by judging the convergence of the series. At the same time, after the 51st Mersenne prime was proved in 2018, a conservative prediction is made, that is, there are at least 52 Mersenne primes in the Mersenne numbers less than 10^215000000; a more accurate prediction is that there are at least 52 Mersenne numbers in the Mersenne numbers less than10^103000000 prime numbers.
Category: Number Theory
[5] viXra:2209.0012 [pdf] submitted on 2022-09-02 01:02:02
Authors: Tae Beom Lee
Comments: 4 Pages.
The Riemann zeta function(RZF) is a function of a complex variable s=x+iy, which is analytic for x>1. The Dirichlet Eta Function(DEF) is also a function of a complex variable s, which is analytic for x>0. The zeros of RZF and DEF are all same. The Riemann hypothesis(RH) states that the non-trivial zeros of RZF is of the form s=0.5+iy. The clue of our proof stems from the symmetry properties of RZF zeros. The two zeros should be on the two edge lines of a strip. But, the parallel two edge lines can’t intersect at the origin, when mapped by DEF. So, RH is true.
Category: Number Theory
[4] viXra:2209.0010 [pdf] submitted on 2022-09-01 13:41:30
Authors: Junho Choi
Comments: 5 Pages.
An equation regarding the density of twin primes had been presented in a preceding research. In this paper, we develop the density equation to explicitly estimate the number of twin primes and propose a conjecture related to the Twin Prime Conjecture based on the method.
Category: Number Theory
[3] viXra:2209.0008 [pdf] replaced on 2023-08-02 23:29:32
Authors: Dante Servi
Comments: 2 Pages.
This article is the presentation of three of my articles with which I have provided graphical proof that the Riemann hypothesis is true. I admit that the previous statement was a forcing, but read the third article to the end.
Category: Number Theory
[2] viXra:2209.0004 [pdf] replaced on 2025-10-23 19:05:23
Authors: Roberto Iannone
Comments: 5 Pages.
In 1994 the mathematician Andrews Wiles proved, using concepts of modern mathematics, namelythe Modular Elliptic Curves, the Fermat's Last Theorem. That demonstration is long and it isunderstandable to mathematic specialists, moreover, these mathematical concepts were not knownat the time when Fermat lived, so he could not prove it throug this road. I thought that there mightbe another simpler proof which would use the properties of algebraic equations and inequalities.These mathematical concepts were known at the time in which Fermat lived and which, therefore,could apply for the proof of the theorem of which he wrote, in the margin of a page of the book ofArithmetica of Diophantus he was reading, to have found a demonstration "wonderful," but that hehad not could to write to the narrowness of the margin itself. I propose, therefore, the followingdemonstration that directly uses the mathematical properties of algebraic equations and inequalitiesthat are understood by all those who study algebra.
Category: Number Theory
[1] viXra:2209.0003 [pdf] replaced on 2024-05-23 22:24:29
Authors: Roberto Iannone
Comments: 5 Pages.
In 1993, the banker Andrew Beal, fond of number theory, analyzing the Fermat's Last Theorem and generalizing it, formulated the conjecture that the exponents of the powers of the equation, the bases of which are coprime, can be of different degree, provided that the degree of one of the powers is equal to 2. The proof of Beal's conjecture which I propose descends consequently by direct demonstration of the Fermat's Last theorem formulated by me, with the use of the mathematical properties of algebraic equations and inequalities.
Category: Number Theory
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