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[33] viXra:1912.0540 [pdf] submitted on 2019-12-31 15:11:41
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper we discuss the Erd\'{o}s-Straus conjecture. Using a very simple method we show that for each $L\in \mathbb{N}$ with $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align}\frac{n}{L}\ll \sum \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{n}{L}\nonumber
\end{align}In particular, for each $L\geq 3$ there exist some $(x_1,x_2,x_3)\in \mathbb{N}^3$ with $x_1\neq x_2$, $x_2\neq x_3$ and $x_3\neq x_1$ such that \begin{align}c_1\frac{3}{L}\leq \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}\leq c_2\frac{3}{L}\nonumber
\end{align}for some $c_1,c_2>1$.
Category: Number Theory
[32] viXra:1912.0538 [pdf] submitted on 2019-12-31 15:30:25
Authors: Theophilus Agama
Comments: 4 Pages.
In this short note we introduce a function which iteratively behaves in a similar fashion compared to the factorial function. However the growth rate of this function is not as dramatic and sudden as the factorial function. We also propose an approximation for this function for any given input, which holds for sufficiently large values of n.
Category: Number Theory
[31] viXra:1912.0531 [pdf] submitted on 2019-12-31 01:10:18
Authors: Kelly Harris
Comments: 4 Pages.
Balancing numbers as introduced by Behera and Panda [1] can be shown to be connected to the formula x^2+1=N in a very simple way. The goal of this paper is to show that if a balancing number exists for the balancing equation 1+ 2+ ... + (y-1) = (y+1)+(y+2)+...+(y+m), then there is a corresponding(2y)^2+1=N, where N is composite. We will also show how this can be used to factor N.
Category: Number Theory
[30] viXra:1912.0528 [pdf] replaced on 2021-08-17 19:52:36
Authors: Theophilus Agama
Comments: 7 Pages. Proof technique greatly simplified with detailed exposition.
In this paper we prove the twin prime conjecture by showing that
\begin{align}
\sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber
\end{align}where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular it follows that
\begin{align}
\sum \limits_{p,p+2\in \mathbb{P}}1=\infty\nonumber
\end{align}by taking $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form
\begin{align}
\sum \limits_{n\leq x}G(n)G(n+l)\nonumber
\end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$.
Category: Number Theory
[29] viXra:1912.0526 [pdf] submitted on 2019-12-31 06:37:40
Authors: Surajit Ghosh
Comments: 34 Pages.
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
Category: Number Theory
[28] viXra:1912.0507 [pdf] submitted on 2019-12-30 06:29:13
Authors: Michele Nardelli, Antonio Nardelli
Comments: 130 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with various parameters of Particle Physics, Dark Matter, Dark Energy and Cosmology
Category: Number Theory
[27] viXra:1912.0494 [pdf] replaced on 2020-01-03 18:42:37
Authors: Toshiro_Takami
Comments: 5 Pages.
I proved the Twin Prime Conjecture.
The probability twin prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in.
I investigated up to 5$\times10^{12}$.\\
When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\
However, it is not 0. Because, primes continue to be produced. Therefore, Twin Primes continue to be produced.\\
If the Twin Primes is finite, the primes is finite.\\
This is because slightly lower than 4/3 times the square of the probability of primes is the probability of Twin Primes.\\
This is contradiction.
Because there are an infinite of primes.\\
\ \\
$[Probability\ of\ the\ Existence\ of\ primes]^2\times4/3$=\\
(Probability\ of\ the\ Existence\ of\ Twin\ Primes)\\
When the number becomes extreme, the generation of prime numbers becomes extremely small. However, it is not 0.\\
Very few, but prime numbers are generated.\\
Therefore, even if the number reaches the limit, twin prime numbers are also generated.\\
That is, Twin Primes exist forever.\\
Category: Number Theory
[26] viXra:1912.0476 [pdf] submitted on 2019-12-27 02:59:52
Authors: Roberto Amato
Comments: 7 Pages. Summary of results extracted from the work [1] R. Amato, A characterization of pythagorean triples, JP Journal of Algebra, Number Theory and Applications, 39, (2017), 221–230
This way has the convenience to find easily all Pythagorean triples x, y, z ∈ N, where x is a predetermined integer, which means finding all right triangles whose sides have integer measures and one cathetus is predetermined.
Category: Number Theory
[25] viXra:1912.0468 [pdf] submitted on 2019-12-27 09:53:57
Authors: Ananthakrishnan S
Comments: 4 Pages.
“It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or
in general any power higher than the second into two powers of like degree; I have
discovered a truly remarkable proof which this margin is too small to contain."
I have discovered the shortest proof by using trigonometric principles.
Category: Number Theory
[24] viXra:1912.0402 [pdf] submitted on 2019-12-22 06:41:15
Authors: Michele Nardelli, Antonio Nardelli
Comments: 101 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described further possible mathematical connections with some parameters of Particle Physics and Cosmology: Pion meson mass (139.57 MeV), Higgs boson mass, scalar meson f0(1710) mass, hypothetical gluino mass and Cosmological Constant value.
Category: Number Theory
[23] viXra:1912.0380 [pdf] submitted on 2019-12-20 11:58:14
Authors: Edoardo Gueglio
Comments: 3 Pages.
CONJECTURES ABOUT THE DIFFERENCE OF THE
SEQUENCE OF RADICALS OF FIRST GRADE POLYNOMIAL
SEQUENCES
Category: Number Theory
[22] viXra:1912.0334 [pdf] submitted on 2019-12-18 04:54:31
Authors: Michele Nardelli, Antonio Nardelli
Comments: 111 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described further possible mathematical connections with some parameters of Particle Physics and Cosmology
Category: Number Theory
[21] viXra:1912.0324 [pdf] submitted on 2019-12-16 22:52:57
Authors: S.Ao, L.Lu
Comments: 25 Pages.
We developed a new Sieve, A-Sieve, and produced some new ways in the study of prime numbers. By using this new method, we can prove that for any natural number k, there are infinitely many pairs of primes that differ by 2k.
Category: Number Theory
[20] viXra:1912.0312 [pdf] replaced on 2019-12-22 03:36:32
Authors: Hassine Saidane
Comments: 1 Page.
The so-called Fermat's last theorem is actually a conjecture that was proposed by Pierre de Fermat in 1637 where he stated that the Diophantine equation xn +yn = zn, with x, y, z and n being positive integers, has no nonzero solution for n > 2. This conjecture was one of the most notable unsolved problems of mathematics. It was finally proven by Wiles and R. Taylor in late 1994 at the cost of long and complex analysis using new mathematical tools that are not analytically accessible to most mathematicians. In this note, we present a short and easy to grasp solution that is based on an optimization approach.
Category: Number Theory
[19] viXra:1912.0303 [pdf] submitted on 2019-12-16 15:20:11
Authors: Edgar Valdebenito
Comments: 1 Page.
We give three integrals
Category: Number Theory
[18] viXra:1912.0302 [pdf] submitted on 2019-12-16 15:21:39
Authors: Edgar Valdebenito
Comments: 2 Pages.
We recall two curious integrals.
Category: Number Theory
[17] viXra:1912.0278 [pdf] submitted on 2019-12-14 13:40:57
Authors: Edgar Valdebenito, Rodrigo Valdebenito
Comments: 6 Pages.
In this note we give a collection of mathematical formulas involving Pi.
Category: Number Theory
[16] viXra:1912.0245 [pdf] replaced on 2019-12-20 11:56:39
Authors: Edoardo Gueglio
Comments: 2 Pages.
CONJECTURES ABOUT THE DIFFERENCE OF THE
SEQUENCE OF RADICALS OF THE NATURAL NUMBERS
Category: Number Theory
[15] viXra:1912.0225 [pdf] submitted on 2019-12-12 05:58:40
Authors: Surajit Ghosh
Comments: 41 Pages.
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined.
Category: Number Theory
[14] viXra:1912.0208 [pdf] submitted on 2019-12-10 23:15:55
Authors: Aaron chau
Comments: 2 Pages.
然而黎曼的假设，挥之不去地刺激了世界各民族的数学家。 但他们在受到了刺激之后，究竟干了一些什么呢？
可惜有些人还在把伪造质数说成是“数学主流”, 并又靠伪造质数来争取学位，难道数学的主流就是伪造质数？
也事实上推翻黎曼假设需要越筒单越有智慧的算术。
Category: Number Theory
[13] viXra:1912.0207 [pdf] replaced on 2019-12-14 15:57:03
Authors: Toshiro Takami
Comments: 4 Pages.
I considered the Twin Prime Conjecture.
The probability twin prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in.\\
When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\
The places where prime numbers come out are filled with multiples of primes one after another, and eventually disappear almost.\\
Primes can only occur very rarely when the numbers are huge.\\
This is natural from the following equation.\\
\begin{equation}
\pi(x)\sim\frac{x}{\log{x}}\ \ \ (x\to\infty)
\end{equation}\\
$[Probability\ of\ the\ Existence\ of\ primes]^2\times4/3\sim$
(Probability\ of\ the\ Existence\ of\ Twin\ Primes)\\
When the number becomes extreme, the generation of primes becomes extremely small. However, it is not 0.\\
Very few, but primes are generated.\\
If the twin primes appears as two primes completely independently, Twin Prime Problem is denied.\\
However, if twin primes appear in combination and appear like primes, twin primes consist forever and Twin Prime Problem is correct.\\
Category: Number Theory
[12] viXra:1912.0205 [pdf] submitted on 2019-12-11 02:48:40
Authors: Toshiro Takami
Comments: 4 Pages.
There are almost no primes in the infinite world.
This is because the place where the primes appears is occupied by multiple of the primes.
If you think about a hexagon, you can see it right away.
Category: Number Theory
[11] viXra:1912.0194 [pdf] submitted on 2019-12-10 04:23:59
Authors: Michele Nardelli, Antonio Nardelli
Comments: 82 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with some parameters of Particle Physics and Cosmology
Category: Number Theory
[10] viXra:1912.0174 [pdf] submitted on 2019-12-08 21:14:39
Authors: Takamasa Noguchi
Comments: 8 Pages.
I made an approximate formula.
In the formula, when N is small, the accuracy is very bad, but as N increases, the accuracy also improves.
Category: Number Theory
[9] viXra:1912.0157 [pdf] replaced on 2019-12-28 23:10:57
Authors: Toshiro Takami
Comments: 5 Pages.
I proved the Twin Prime Conjecture.
The probability that (6n -1) is a prime and (6n+1) is also a prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in.
I investigated up to 5$\times10^{12}$.\\
All Twin Primes are produced in hexagonal circulation. It does not change in a huge number (forever huge number).\\
The production of Twin Primes equal the existence of Twin Primes.\\
When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\
However, it is not 0. Because, primes continue to be produced. Therefore, Twin Primes continue to be produced.\\
If the Twin Primes is finite, the primes is finite.\\
This is because slightly lower than 4/3 times the square of the probability of primes is the probability of Twin Primes.
This is contradiction.
Because there are an infinite of primes.\\
Category: Number Theory
[8] viXra:1912.0153 [pdf] replaced on 2019-12-08 07:55:05
Authors:
Comments: Pages. Correction to first version (v1) is to replace COS with SIN in the third vector, marked as (3)
No zeta zero is off the critical line. Riemann Hypothesis is false if and only if geometrical objects A, B and C are coplanar. The proof shows that A, B and C are linearly independent, so not coplanar.
Category: Number Theory
[7] viXra:1912.0151 [pdf] submitted on 2019-12-08 01:15:38
Authors: Toshiro Takami
Comments: 8 Pages.
If (p,p+2) are twin primes, (p+30, p+2+30) or (p+60, p+2+60) or (p+90, p+2+90) or (p+120, p+2+120) or (p+150, p+2+150) ) or (p+180, p+2+180) or (p+210, p+2+210) or (p+240, p+2+240)……. is to be a twin primes.\\
There are three type of twin primes, last numbers are (1, 3)..(7, 9)..(9, 1).\\
They are lined up at intervals such as 30 or 60 or 90 or 120 or 150 or 180 or 210 or 240 or 270 or 300 etc.\ That is, it is a multiple of 30. \\
Repeat this.\\
And the knowledge about prime numbers is also taken into account.\\
That is, Twin Primes exist forever.\\
Category: Number Theory
[6] viXra:1912.0148 [pdf] submitted on 2019-12-08 06:34:20
Authors: Michele Nardelli, Antonio Nardelli
Comments: 147 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with some sectors of Particle Physics (values of Pion mesons and other baryons and mesons) and Cosmology
Category: Number Theory
[5] viXra:1912.0119 [pdf] submitted on 2019-12-06 02:07:40
Authors: Takamasa Noguchi
Comments: 4 Pages. Explanation of effective prime number judgment method even for Carmichael number.
Explanation of effective prime number judgment method even for Carmichael number.
This method of judgment does not give a 100% correct answer.
Care must be taken especially for (n=p^k (P=Prime)) with primitive roots.
Category: Number Theory
[4] viXra:1912.0117 [pdf] submitted on 2019-12-06 04:20:47
Authors: Matanari Shimoinuda
Comments: 32 Pages.
In this article, we review the local Langlands program. I hope that this article is a useful guide to understand Langlands program.
Category: Number Theory
[3] viXra:1912.0108 [pdf] submitted on 2019-12-06 06:49:41
Authors: Michele Nardelli, Antonio Nardelli
Comments: 83 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with some sectors of Particle Physics and Cosmology
Category: Number Theory
[2] viXra:1912.0087 [pdf] submitted on 2019-12-05 05:25:17
Authors: Michele Nardelli, Antonio Nardelli
Comments: 133 Pages.
In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with some sectors of Particle Physics and Cosmology
Category: Number Theory
[1] viXra:1912.0028 [pdf] submitted on 2019-12-02 13:53:35
Authors: BOUARFAOUI Saïd
Comments: 8 Pages. Some studies on the 3n+1 problem
In the present paper, we study the 3n+1 problem to know if there is a theoretical number such as the number never land on the cycle {4;2;1}and grows continually.We will specify the general formula such as the number of uneven steps (3��+1)and even steps (��/2)are equals.We want to know also the uneven number form 2^��+2^0 and its composition during ��uneven (3��+1)and even steps (��/2).The composition number 2��+20will be viewed in two ways :1. When the powers which compose the numbers are expanded. We will recognize the Pascal’s triangle.2. When the powers which compose the numbers are agglutinate. We will recognize (3^��) in base 2.
Category: Number Theory
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