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2019 - 1901(11) - 1902(11) - 1903(21) - 1904(25) - 1905(22) - 1906(38) - 1907(42) - 1908(20) - 1909(34) - 1910(48) - 1911(29) - 1912(36)

2020 - 2001(25)

Any replacements are listed farther down

[2251] **viXra:2001.0372 [pdf]**
*submitted on 2020-01-19 17:41:34*

**Authors:** Nikos Mantzakouras

**Comments:** 34 Pages.

The Beal conjecture is a number theory formulated in 1993 by the billionaire banker, Mr Andrew Beal.
Mr Beal, very recently, declared a one-million-dollar award for the proof of this number theory. As at present, no proof of this conjecture has been generally found. In this article, we provided the proof for Beal conjecture in a crystal-clear systematic approach.Proof implies that we know the proof of Fermat's last theorem.

**Category:** Number Theory

[2250] **viXra:2001.0363 [pdf]**
*submitted on 2020-01-19 08:01:43*

**Authors:** Algirdas Antano Maknickas

**Comments:** 1 Page.

These short remarks show deriviation of Birch and Swinnerton-Dyer conjungture. As a consequence new one resulting constant free equality of Birch and Swinnerton-Dyer conjungture proposed

**Category:** Number Theory

[2249] **viXra:2001.0362 [pdf]**
*submitted on 2020-01-19 08:27:33*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 94 Pages.

In this research thesis, we have analyzed and deepened further Ramanujan expressions applied to some sectors of String Theory and Particle Physics. We have therefore described new possible mathematical connections.

**Category:** Number Theory

[2248] **viXra:2001.0309 [pdf]**
*submitted on 2020-01-16 03:00:28*

**Authors:** Nikos Mantzakouras

**Comments:** 7 Pages.

Twin prime conjecture, also known as Polignac’s conjecture, in number theory, assertion that there are infinitely many twin primes, or pairs of primes that differ by 2. The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, who wrote that any even number can be expressed in infinite ways as the difference between two consecutive primes

**Category:** Number Theory

[2247] **viXra:2001.0299 [pdf]**
*submitted on 2020-01-16 09:19:51*

**Authors:** Kouji Takaki

**Comments:** 12 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[2246] **viXra:2001.0273 [pdf]**
*submitted on 2020-01-14 23:53:47*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Contains Rigorous Proofs for Riemann Hypothesis (and Explanations for two types of Gram points), Polignac's and Twin Prime Conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first key step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second key step consisting of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[2245] **viXra:2001.0265 [pdf]**
*submitted on 2020-01-15 06:53:40*

**Authors:** Nikos Mantzakouras

**Comments:** 8 Pages.

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES.

**Category:** Number Theory

[2244] **viXra:2001.0255 [pdf]**
*submitted on 2020-01-14 12:28:33*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 115 Pages.

In this research thesis, we have analyzed and deepened some equations concerning the Ramanujan continued fractions. We have described further possible mathematical connections with some parameters of Particle Physics and Cosmology.

**Category:** Number Theory

[2243] **viXra:2001.0253 [pdf]**
*submitted on 2020-01-14 13:03:56*

**Authors:** Nikos Mantzakouras

**Comments:** 20 Pages. In International Conference From Nina Ringo on Mathematics and Mechanics 16 May 2018

Abstract: The Riemann zeta function is one of the most Euler’s important and fascinating functions in mathematics. By analyzing the material of Riemann’s conjecture, we divide our analysis in the ζ(z) function and in the proof of the conjecture, which has very important consequences on the distribution of prime numbers. The proof of the Hypothesis of Riemann result from the simple logic, that when two properties are associated, (the resulting equations that based in two Functional equations of Riemann ), if zero these equations, ie ζ(z) = ζ (1-z)= 0 and simultaneously they to have the proved property 1-1 of the Riemann function ζ(z).Thus, there is not margin for to non exist the Re (z) = 1/2 {because ζ (z) = ζ (1-z)=0 and also ζ(z) as and ζ(1-z) are 1-1}.This, as it stands, will gives the direction of all the non-trivial roots to be all in on the critical line, with a value in the real axis equal 1/2.

**Category:** Number Theory

[2242] **viXra:2001.0224 [pdf]**
*submitted on 2020-01-13 08:15:10*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 82 Pages.

In this research thesis, we have analyzed and deepened some equations concerning the Ramanujan continued fractions. Further possible mathematical connections with some parameters of Particle Physics and Cosmology.

**Category:** Number Theory

[2241] **viXra:2001.0204 [pdf]**
*submitted on 2020-01-11 17:12:48*

**Authors:** Theophilus Agama

**Comments:** 7 Pages.

In this paper we introduce and develop the theory of the Collatz process. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie germain primes. We also provide several formulation of the Collatz conjecture in this language.

**Category:** Number Theory

[2240] **viXra:2001.0179 [pdf]**
*submitted on 2020-01-10 10:06:52*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

In this research thesis, we continue to analyze and deepen further Ramanujan’s equations of Manuscript Book 2 and describe new possible mathematical connections with some parameters of Particle Physics and Cosmology.

**Category:** Number Theory

[2239] **viXra:2001.0152 [pdf]**
*submitted on 2020-01-09 04:52:48*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. Submitted to the journal Functiones & Approximatio Commentari Mathematici

In this paper, assuming that $c

**Category:** Number Theory

[2238] **viXra:2001.0151 [pdf]**
*submitted on 2020-01-09 04:58:51*

**Authors:** Surajit Ghosh

**Comments:** 40 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

[2237] **viXra:2001.0144 [pdf]**
*submitted on 2020-01-09 05:52:45*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

In this research thesis, we continue to analyze and deepen further Ramanujan’s equations of Manuscript Book 2 and described new possible mathematical connections with some parameters of Particle Physics and Black Holes Physics.

**Category:** Number Theory

[2236] **viXra:2001.0130 [pdf]**
*submitted on 2020-01-08 19:43:35*

**Authors:** OttoAltorfer

**Comments:** 7 Pages.

A introdução de uma função de números inédita facilitou a solução do último Teorema de Fermat.

**Category:** Number Theory

[2235] **viXra:2001.0116 [pdf]**
*submitted on 2020-01-07 10:56:57*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 87 Pages. UPDATED VERSION

In this research thesis, we have analyzed further formulas of Manuscript Book 1 of Srinivasa Ramanujan and described new possible mathematical connections with various parameters of Particle Physics and Cosmology (Cosmological Constant, some parameters of Dark Energy)

**Category:** Number Theory

[2234] **viXra:2001.0113 [pdf]**
*submitted on 2020-01-07 12:56:46*

**Authors:** Richard L. Hudson

**Comments:** 7 Pages.

Originated by Lothar Collatz in 1937 [1], the conjecture states: given the recursive function, y=3x+1 if x is odd, or y=x/2 if x is even, for any positive integer x, y will equal 1 after a finite number of steps. This analysis examines number form and uses a tree type graph to prove the process.

**Category:** Number Theory

[2233] **viXra:2001.0097 [pdf]**
*submitted on 2020-01-06 14:00:37*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 11 Pages. Submitted to the journal Inventiones Mathematicae

In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof that $c<3rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=\frac{3}{e}.e^{\left(\frac{1}{\epsilon^2} \right)}$ for $0<\epsilon<1$ and $K(\epsilon)=3$ for $\epsilon \geq1$. Some numerical examples are presented.

**Category:** Number Theory

[2232] **viXra:2001.0084 [pdf]**
*submitted on 2020-01-06 00:20:00*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of a modiﬁed sieve of Eratosthenes to prove the Goldbach conjecture.

**Category:** Number Theory

[2231] **viXra:2001.0072 [pdf]**
*submitted on 2020-01-05 13:46:30*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 119 Pages.

In this research thesis, we have analyzed further formulas of Manuscript Book 1 of Srinivasa Ramanujan and described new possible mathematical connections with various parameters of Particle Physics and Cosmology (Cosmological Constant, some parameters of Dark Energy)

**Category:** Number Theory

[2230] **viXra:2001.0069 [pdf]**
*submitted on 2020-01-05 17:30:59*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of a modiﬁed sieve of Eratosthenes to prove the twin prime conjecture.

**Category:** Number Theory

[2229] **viXra:2001.0056 [pdf]**
*submitted on 2020-01-04 11:52:12*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 105 Pages.

In this research thesis, we have analyzed further Ramanujan formulas and described new possible mathematical connections with various parameters of Particle Physics and Cosmology

**Category:** Number Theory

[2228] **viXra:2001.0005 [pdf]**
*submitted on 2020-01-01 15:38:39*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 130 Pages.

In this research thesis, we have analyzed further Ramanujan formulas inherent Highly composite numbers and described new possible mathematical connections with various parameters of Particle Physics, Dark Matter, Dark Energy and Cosmology

**Category:** Number Theory

[2227] **viXra:2001.0004 [pdf]**
*submitted on 2020-01-01 17:56:45*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2020 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two equations for n ≥ 3 and n ≥ 2, with both not tautologous, and hence refuting the Erdős-Strauss conjecture. These form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[2226] **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

[2225] **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

[2224] **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

[2223] **viXra:1912.0528 [pdf]**
*submitted on 2019-12-31 05:47:38*

**Authors:** Theophilus Agama

**Comments:** 5 Pages.

In this paper we prove the twin prime conjecture by showing that \begin{align}\sum \limits_{n\leq x}\Lambda(n)\Lambda(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber
\end{align}for some $\mathcal{C}:=\mathcal{C}(2)>0$. 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

[2222] **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

[2221] **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

[2220] **viXra:1912.0494 [pdf]**
*submitted on 2019-12-28 11:07:51*

**Authors:** Toshiro_Takami

**Comments:** 14 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

[2219] **viXra:1912.0488 [pdf]**
*submitted on 2019-12-27 22:12:45*

**Authors:** Jose R. Sousa

**Comments:** 28 Pages. This is a cleaned up and improved version, the original is on arXiv

This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$.
I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $\psi$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers.
We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $\zeta(2k+1)$ and a new generating function for $\zeta(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

**Category:** Number Theory

[2218] **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

[2217] **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

[2216] **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

[2215] **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

[2214] **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

[2213] **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

[2212] **viXra:1912.0312 [pdf]**
*submitted on 2019-12-16 11:02:37*

**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 regarding the Diophantine equation xn +yn = zn, where x, y, z and n are integers, having 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 accessible to the common mathematician. In this note, we present a short and accessible solution to all based on an optimization approach.

**Category:** Number Theory

[2211] **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

[2210] **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

[2209] **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

[2208] **viXra:1912.0245 [pdf]**
*submitted on 2019-12-13 06:04:57*

**Authors:** Edoardo Gueglio

**Comments:** 2 Pages.

CONJECTURES ABOUT THE DIFFERENCE OF THE
SEQUENCE OF RADICALS OF THE NATURAL NUMBERS

**Category:** Number Theory

[2207] **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

[2206] **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

[2205] **viXra:1912.0207 [pdf]**
*submitted on 2019-12-11 01:03:26*

**Authors:** Toshiro Takami

**Comments:** 9 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

[2204] **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

[2203] **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

[2202] **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

[2201] **viXra:1912.0157 [pdf]**
*submitted on 2019-12-08 17:27:50*

**Authors:** Toshiro Takami

**Comments:** 15 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

[2200] **viXra:1912.0153 [pdf]**
*submitted on 2019-12-08 00:02:55*

**Authors:**

**Comments:** 1 Page.

No zero is off the critical line.
Riemann Hypothesis is true iff A, B and C are coplanar in infinite dimensional space.
A and B are geometrical objects directly related to (supposed symmetric distinct) zeta roots and C is derived from A and B.
Proof shows A, B and C can not be coplanar (unless $\delta$ is zero).

**Category:** Number Theory

[2199] **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

[2198] **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

[2197] **viXra:1912.0124 [pdf]**
*submitted on 2019-12-06 14:34:36*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:
”Every even integer greater than 2 can be expressed as the sum of two primes”.
Manuscript content: Prove that Goldbach’s conjecture is correct.
Key words: Prime numbers, Goldbach’s conjecture, number theory.

**Category:** Number Theory

[2196] **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

[2195] **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

[2194] **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

[2193] **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

[2192] **viXra:1912.0082 [pdf]**
*submitted on 2019-12-03 20:13:14*

**Authors:** John Yuk Ching Ting

**Comments:** 49 Pages. Reverse-Engineered Rigorous Proofs for Riemann Hypothesis (and Explanations for two types of Gram points), Polignac's and Twin Prime Conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered rigorous proofs and explanations.

**Category:** Number Theory

[2191] **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

[2190] **viXra:1911.0525 [pdf]**
*submitted on 2019-11-30 10:17:22*

**Authors:** Miguel Cerdá Bennassar

**Comments:** 7 Pages.

Este escrito es continuación del publicado en Agosto 2019 con título "LA CONJETURA DE COLLATZ. Orden y armonía en los números de las secuencias".

**Category:** Number Theory

[2189] **viXra:1911.0484 [pdf]**
*submitted on 2019-11-29 09:47:14*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

Andrew Wiles proved there are no integers x, y, and z and a prime p ≥ 3 with x^p+y^p +z^p=0.
We use the Barlow relations to generate Fermat Triples where x^p+y^p +z^p ≡ 0 for an infinite number of moduli.

**Category:** Number Theory

[2188] **viXra:1911.0477 [pdf]**
*submitted on 2019-11-28 02:22:12*

**Authors:** Dmitri Martila

**Comments:** 3 Pages.

Due to the Incompleteness Theorems of Gödel one can say, that some true conjectures do not have valid proofs. One could think it also about my conjectures below, but I was lucky to find evidence for them.

**Category:** Number Theory

[2187] **viXra:1911.0426 [pdf]**
*submitted on 2019-11-25 06:34:15*

**Authors:** Juan Moreno Borrallo

**Comments:** 4 Pages.

In this paper it is proposed a nice rational estimator of the fractional part of the square root of any positive integer.

**Category:** Number Theory

[2186] **viXra:1911.0316 [pdf]**
*submitted on 2019-11-18 11:12:58*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

In this paper it is proved that the sum of consecutive prime numbers up to the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, they are found some solutions such that both series are equal. Finally, they are listed the prime numbers at which both series are equal, and exposed some conjectures regarding this type of prime numbers.

**Category:** Number Theory

[2185] **viXra:1911.0310 [pdf]**
*submitted on 2019-11-18 05:46:30*

**Authors:** Aaron chau

**Comments:** 2 Pages.

在西方的古希腊，Euclid 证明质数无限，他是用(乘除法)来表述反证法；
而现时在东方香港，本文同时来证明孪生质数无限，黎曼假设被推翻； 筆者是用(加减法)来表述永恒的多与少。

**Category:** Number Theory

[2184] **viXra:1911.0308 [pdf]**
*submitted on 2019-11-18 07:05:36*

**Authors:** James Edwin Rock

**Comments:** 2 Pages.

In the 17th century Christian Goldbach conjectured that any even number four or greater is the sum of two primes. This has never been proven, but it has been tested and shown to be true for all even numbers up to 4 x 10^18. We show the probability of this being false is about (1 / 10^98912 ) collectively for all even integers greater than 65,536,000 and drops to over ((1 / 10)^10)^14 collectively for all even integers greater than 4.5 x 10^18.

**Category:** Number Theory

[2183] **viXra:1911.0287 [pdf]**
*submitted on 2019-11-17 08:18:55*

**Authors:** Richard J. Mathar

**Comments:** 39 Pages.

The Number-of-Divisors Function tau(n) is the number of divisors of a positive integer n,
including 1 and n itself.
Searching for pairs of the format (tau(n), tau(n+1)), some pairs appear (very) often, some never
and some --- like (1,2), (4,9), or (10,3) --- exactly once. The manuscript provides proofs for 46 pairs to appear exactly once
and lists 12 pairs that conjecturally appear only once.
It documents a snapshot of a community effort to verify sequence A161460 of the Online
Encyclopedia of Integer Sequences that started ten years ago.

**Category:** Number Theory

[2182] **viXra:1911.0236 [pdf]**
*submitted on 2019-11-13 14:02:48*

**Authors:** David Streit, Christoph Benzmüller, Ralf Wüsthofen

**Comments:** 35 Pages.

This document presents the discussion with two experts in automated theorem proving in the course of drawing up their technical report http://viXra.org/abs/1910.0115 on the paper 'The Inconsistency of Arithmetic' (http://vixra.org/abs/1904.0428). Main issue of the discussion is the wrong countermodel given in the report that is based on an incorrect application of predicate logic.

**Category:** Number Theory

[2181] **viXra:1911.0208 [pdf]**
*submitted on 2019-11-11 14:09:39*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the AAFrempong conjecture, named by the author after the name of the author, which states in any triangle, the sum of the lengths of any two sides is greater than the length of the third side, is not tautologous. This forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[2180] **viXra:1911.0201 [pdf]**
*submitted on 2019-11-11 02:24:22*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Prime Triplet and Twin Primes have exactly the same dynamics.
All Prime Triplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Triplet are generated only at (6n -1)(6n+1). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Triplet are 35/12 times of the 3th power distribution of primes, the frequency of occurrence of Prime Triplet is very equal to 0.
However, it is not 0. Therefore, Prime Triplet continue to be generated.
If Prime Triplet is finite, the Primes is finite.
The probability of Prime Triplet 35/12 times of the 3th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Triplet exist forever.

**Category:** Number Theory

[2179] **viXra:1911.0198 [pdf]**
*submitted on 2019-11-11 07:34:34*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 172 Pages.

In this research thesis, we have analyzed some parts of Ramanujan’s Manuscripts and obtained new mathematical connections between several Ramanujan’s equations, the Rogers-Ramanujan continued fractions and some sectors of Cosmology and Theoretical Physics

**Category:** Number Theory

[2178] **viXra:1911.0180 [pdf]**
*submitted on 2019-11-09 16:17:51*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Sextuplet and Twin Primes have exactly the same dynamics.
All Prime Sextuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Sextuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Sextuplet are 48/3 times of the sixth power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Sextuplet continue to be generated.
If Prime Sextuplet is finite, the Primes is finite.
The probability of Prime Sextuplet 48/3 times of the sixth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Sextuplet exist forever.

**Category:** Number Theory

[2177] **viXra:1911.0179 [pdf]**
*submitted on 2019-11-09 17:24:44*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Prime Quintuplet and Twin Primes have exactly the same dynamics.
All Prime Quintuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quintuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quintuplet are 96/3 times of the 5th power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Quintuplet continue to be generated.
If Prime Quintuplet is finite, the Primes is finite.
The probability of Prime Quintuplet 96/3 times of the 5th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quintuplet exist forever.

**Category:** Number Theory

[2176] **viXra:1911.0177 [pdf]**
*submitted on 2019-11-09 01:24:48*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

Sexy Primes Conjecture were prooved.
Sexy Primes and Twin Primes and Cousin Primes have exactly the same dynamics.
All Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Sexy Primes are generated only at (6n+1)(6n -1). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Sexy Primes are 8/3 times the square of the distribution of primes, the frequency of occurrence of Sexy Primes is very equal to 0.
However, it is not 0. Therefore, Sexy Primes continue to be generated.
If Sexy Primes is finite, the Primes is finite.
Because, Sexy Primes are 8/3 times the square of the distribution of primes.
This is contradictory. Since there are an infinite of Primes.
That is, Sexy Primes exist forever.

**Category:** Number Theory

[2175] **viXra:1911.0144 [pdf]**
*submitted on 2019-11-08 07:33:56*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Quadruplet and Twin Primes have exactly the same dynamics.
All Prime Quadruplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quadruplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quadruplet are 16/3 of the fourth power distribution of primes, the frequency of occurrence of Prime Quadruplet is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated.
If Prime Quadruplet is finite, the Primes is finite.
The probability of Prime Quadruplet 16/3 of the fourth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quadruplet exist forever.

**Category:** Number Theory

[2174] **viXra:1911.0116 [pdf]**
*submitted on 2019-11-06 12:19:15*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 117 Pages.

In this research thesis, we have analyzed some parts of Ramanujan’s Manuscripts and obtained new mathematical connections between several Ramanujan’s equations, the Rogers-Ramanujan continued fractions and the Dilaton value.

**Category:** Number Theory

[2173] **viXra:1911.0111 [pdf]**
*submitted on 2019-11-06 01:41:19*

**Authors:** Wei Zhang

**Comments:** 6 Pages.

In number theory, for the continuous product formula Π(1-2/p), the meaning is unclear.This paper gives the definition and nature of Φ(m) function, as well as the relationship between Φ(m) and Euler’s totient function φ(m). In number theory, Euler function φ(m) is widely used, Φ(m) function if there are other applications, Some attempts are made in this paper.

**Category:** Number Theory

[2172] **viXra:1911.0095 [pdf]**
*submitted on 2019-11-06 10:12:07*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the algorithm used for the sieve of Eratosthenes which is not tautologous, hence refuting the conjecture of incompletely predictable problems such as the Riemann zeta function. We also note that the mode of inference used throughout the source is induction. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[2171] **viXra:1911.0093 [pdf]**
*submitted on 2019-11-05 14:55:47*

**Authors:** John Yuk Ching Ting

**Comments:** 39 Pages. Combined proofs for Riemann hypothesis, Polignac's and Twin prime conjectures with also explanations for two types of Gram points.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Valid and complete mathematical arguments for first key step of converting this function into its continuous format version and second key step of applying Information-Complexity conservation to this Sieve are provided. Direct spin-offs from first step consist of proving Riemann hypothesis (and explaining both types of Gram points) and second step consist of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[2170] **viXra:1911.0083 [pdf]**
*submitted on 2019-11-05 04:38:15*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Cousin Primes Conjecture were performed using WolframAlpha and Wolfram cloud from the beginning this time, as in the case of the twin primes that we did the other day.
Cousin Primes and Twin Primes have exactly the same dynamics.
All Cousin Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Cousin Primes are generated only at (6n+1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Cousin Primes are 4/3 times the square of the distribution of primes, the frequency of occurrence of Cousin Primes is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated. That is, Cousin Primes exist forever.

**Category:** Number Theory

[2169] **viXra:1911.0079 [pdf]**
*submitted on 2019-11-05 09:08:26*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 91 Pages.

In this research thesis, we have described some new possible mathematical connections between various equations concerning the Ramanujan’s sum of two cubes, ζ(2), π, ϕ, Ramanujan’s mock theta functions and some sectors of Theoretical Physics

**Category:** Number Theory

[2168] **viXra:1911.0061 [pdf]**
*submitted on 2019-11-04 01:47:31*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series part3.

**Category:** Number Theory

[2167] **viXra:1911.0052 [pdf]**
*submitted on 2019-11-03 01:40:08*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series.

**Category:** Number Theory

[2166] **viXra:1911.0049 [pdf]**
*submitted on 2019-11-03 02:40:36*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series part2.

**Category:** Number Theory

[2165] **viXra:1911.0044 [pdf]**
*submitted on 2019-11-03 07:05:12*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

In this research thesis, we have described the new possible mathematical connections between the following fundamental mathematical constants: π, ϕ, ζ(2), ζ(6), ζ(8) and ζ(10)

**Category:** Number Theory

[2164] **viXra:1911.0020 [pdf]**
*submitted on 2019-11-01 16:12:52*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 35 Pages.

In the present research thesis, we have obtained further interesting new possible mathematical connections concerning some sectors of Ramanujan’s mathematics, some sectors of Particle Physics, inherent principally the Dark Matter candidate particles and the physics of black holes (Ramanujan-Nardelli mock formula).

**Category:** Number Theory

[2163] **viXra:1911.0019 [pdf]**
*submitted on 2019-11-01 17:30:43*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Riemann hypothesis.

**Category:** Number Theory

[2162] **viXra:1911.0002 [pdf]**
*submitted on 2019-11-01 09:10:26*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

I proved the Twin Prime Conjecture.
However, a new problem of mystery with a Constance 4/3 occurred.
I have studied this in various ways, but I don't know.

**Category:** Number Theory

[2161] **viXra:1910.0654 [pdf]**
*submitted on 2019-10-31 21:02:38*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Goldbach's conjecture.

**Category:** Number Theory

[2160] **viXra:1910.0650 [pdf]**
*submitted on 2019-10-31 03:23:41*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is proof of a famous formula.

**Category:** Number Theory

[2159] **viXra:1910.0636 [pdf]**
*submitted on 2019-10-30 22:53:32*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration3.

**Category:** Number Theory

[2158] **viXra:1910.0634 [pdf]**
*submitted on 2019-10-30 00:30:29*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration２.

**Category:** Number Theory

[2157] **viXra:1910.0565 [pdf]**
*submitted on 2019-10-27 23:13:37*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a collaboration of "ON PRIME NUMBERS" and "Definition".

**Category:** Number Theory

[2156] **viXra:1910.0563 [pdf]**
*submitted on 2019-10-27 02:56:14*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

Adding to the known partial results, two famous Math problems : Beal conjecture and Fermat - Catalan conjecture are proved by one theorem -QS theorem that we propose in this article, and also means that the elementary proof of FLt has been found.

**Category:** Number Theory

[2155] **viXra:1910.0558 [pdf]**
*submitted on 2019-10-27 07:27:30*

**Authors:** William F. Gilreath

**Comments:** 10 Pages. Published in the General Science Journal

Three fallacies that illustrate why division by zero is frequently considered undefined operation are examined. The example fallacies consider the unique case of zero divided by zero. Two examples are fallacies of equality, and the other is an example of ambiguity in the solution for an equation. These fallacies are examined using the transmathematic number nullity F. By utilizing nullity, division by zero is no longer an undefined or indeterminate operation, but a consistent, well-defined operation in arithmetic.

**Category:** Number Theory

[2154] **viXra:1910.0551 [pdf]**
*submitted on 2019-10-27 12:15:29*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

With a strange and ironic twist an open number theory problem, show Zeta(n) is irrational for natural numbers greater than or equal to 2, is solved with the easiest of number theory concepts: the rules of representing fractions with decimals.

**Category:** Number Theory

[2153] **viXra:1910.0549 [pdf]**
*submitted on 2019-10-26 17:08:03*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 108 Pages.

In this research thesis, we have described some new mathematical connections between some equations of various topics concerning the Dilaton value, the D-Brane, the Bouncing Cosmology and some sectors of Number Theory (Riemann’s functions of S. Ramanujan and Rogers-Ramanujan continued fractions).

**Category:** Number Theory

[2152] **viXra:1910.0538 [pdf]**
*submitted on 2019-10-26 07:33:48*

[2151] **viXra:1910.0499 [pdf]**
*submitted on 2019-10-24 02:15:04*

[2150] **viXra:1910.0494 [pdf]**
*submitted on 2019-10-24 05:10:18*

**Authors:** Zhiping Dai

**Comments:** 7 Pages.

Since the set of AS(+) and AS(×) is a bijective function, we use the improved the theorem of asymptotic density to prove that there exist prodcut of two odd primes in any AS(×).
At the same time, in any AS(+), the sum of two odd primes can be obtained.

**Category:** Number Theory

[2149] **viXra:1910.0475 [pdf]**
*submitted on 2019-10-23 21:44:14*

**Authors:** Derek Tucker

**Comments:** 1 Page. Replaces the previous submission

Twin prime conjecture is proven from the observation that all composite odd numbers with factors greater than three occur in the cycle (0pm, 1pm, 5pm, 6pm), This draws circles with diameter 2p_m^2, and inter circle interval of 4p_m^2. For exclusively composite numbers we have |p_m^2±6p_m |.

**Category:** Number Theory

[2148] **viXra:1910.0444 [pdf]**
*submitted on 2019-10-23 12:21:40*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 152 Pages.

In this research thesis, we have described some new mathematical connections between some equations of the Ramanujan’s manuscripts, the Rogers-Ramanujan continued fractions and some sectors of Particle Physics (physical parameters of mesons and dilatons, in particular the values of the masses), String Theory and D-branes.

**Category:** Number Theory

[2147] **viXra:1910.0411 [pdf]**
*submitted on 2019-10-21 03:47:35*

[2146] **viXra:1910.0395 [pdf]**
*submitted on 2019-10-21 11:38:49*

**Authors:** Siddharth Bhatt

**Comments:** 1 Page.

When working with fractions, gravity always acts towards the division bar. This leads to a very non-intuitive result when yeeting a coefficient into the index. Since inverse yeeting is now done along the direction of gravity, the number itself gets inverted after reaching the index.

**Category:** Number Theory

[2145] **viXra:1910.0367 [pdf]**
*submitted on 2019-10-20 11:45:58*

**Authors:** Derek Tucker

**Comments:** 3 Pages.

Proof of Legendre's conjecture by elementary means.

**Category:** Number Theory

[2144] **viXra:1910.0366 [pdf]**
*submitted on 2019-10-20 11:58:01*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 53 Pages. Last version after correcting some topos errors. Submitted to the journal Compositio Mathematica.

In 1997, Andrew Beal announced the following conjecture: Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor. We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or not a common factor. Three numerical examples are given.

**Category:** Number Theory

[2143] **viXra:1910.0365 [pdf]**
*submitted on 2019-10-20 11:57:05*

**Authors:** Derek Tucker

**Comments:** 1 Page.

Let y = exp(ln # - ln x) mod 1. The results show y = 0 for integer x if and only if x is a factor of #.

**Category:** Number Theory

[2142] **viXra:1910.0364 [pdf]**
*submitted on 2019-10-19 15:17:19*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

In this research thesis, we have described some new mathematical connections between some equations of various sectors concerning the D-Branes and some Ramanujan’s modular equations and approximations to π.

**Category:** Number Theory

[2141] **viXra:1910.0349 [pdf]**
*submitted on 2019-10-19 07:42:17*

**Authors:** Y.Mieno

**Comments:** 3 Pages.

A progression and the periodic sequences of the progressions of this kind.

**Category:** Number Theory

[2140] **viXra:1910.0322 [pdf]**
*submitted on 2019-10-18 12:33:20*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 201 Pages.

In this research thesis, we have described some new mathematical connections between some equations of various topics concerning the D-Branes and some sectors of Number Theory (Rogers-Ramanujan continued fractions and mock theta functions).

**Category:** Number Theory

[2139] **viXra:1910.0317 [pdf]**
*submitted on 2019-10-17 02:18:31*

**Authors:** Wu Ye TangYin

**Comments:** 13 Pages. Please forgive me for my low level of mathematics writing. The article is right

According to the random theory and hypothesis theory, the calculation of any number is pushed to infinity. In this paper, 2n-a = 2 * B (B does not know whether it is prime number, or compound number. So the hypothesis plays an important role in judgment. If B is equal to prime, then there is no need to calculate. If B is a compound number, its factorization prime factor, we can get the prime number, and then we can calculate it. But infinity belongs to the unknown.
We don't know what it is to decompose prime factors. Only a, B, C, D.. Then suppose it is a composite number. In this paper, it is only for infinite odd numbers. Is there an inverse column? Odd numbers are not equal to two same prime numbers, plus the sum of odd prime numbers.)

**Category:** Number Theory

[2138] **viXra:1910.0316 [pdf]**
*submitted on 2019-10-17 02:46:52*

**Authors:** Wu Ye TangYin

**Comments:** 13 Pages. Who is willing to help me revise the article? My writing level is low. But theoretical logic is right. Help me get in touch with math magazines.

o prove the idea, assuming that any even number can not be equal to the sum of two prime numbers, then according to the analog computing logic.
Subject: Use hypothesis to judge unknown. In infinite even numbers, there are only numbers, a, b, c, D. Can only be judged; it's prime, or compound.
When: 2N－P=B (B, it is a prime number, or it is a compound number), the hypothesis is used as the basis of judgment. If B equals a prime number, there is no need to calculate it. But B is an unknown number. Judge it to be a prime or a compound number. Assuming a complex number, it can decompose the prime factor. We can get prime numbers. Here, we use hypothesis computing theory to push the unknown to infinity.。Find any even number, there are prime pairs.（Abbreviation：2N=Pa＋Pb）

**Category:** Number Theory

[2137] **viXra:1910.0281 [pdf]**
*submitted on 2019-10-16 00:11:18*

**Authors:** Felix M. Lev

**Comments:** 9 Pages.

The {\it technique} of classical mathematics involves only potential infinity, i.e. infinity is understood only as a limit. However, {\it the basis} of classical mathematics does involve actual infinity: the infinite ring of integers $Z$ is the starting point for constructing infinite sets with different cardinalities, and it is not even posed a problem whether $Z$ can be treated as a limit of finite sets. On the other hand, finite mathematics starts
from the ring $R_p=(0,1,...p-1)$ (where all operations are modulo $p$) and the theory contains only finite sets. We prove that $Z$ can be treated as a limit of $R_p$ when $p\to\infty$ and explain that, as a consequence, finite mathematics is more fundamental than
classical one.

**Category:** Number Theory

[2136] **viXra:1910.0261 [pdf]**
*submitted on 2019-10-15 19:05:23*

**Authors:** Derek Tucker

**Comments:** 7 Pages.

Our objective is to demistify prime gaps in the integers. We will show that the explicit range of prime gaps in the integers is bounded from below by two and above by the expression 〖2p〗_(n-1) , valid for gaps beginning 〖(p〗_n^2-1)-p_(n-1). This upper bound theoretically becomes necessarily greater than empirical observation within empirically verified range, enabling explicit closure on prime gap issues. These results confirm the prime pattens conjecture and the Prime Inter-Square Conjecture (PISC) Legendre’s conjecture.

**Category:** Number Theory

[2135] **viXra:1910.0239 [pdf]**
*submitted on 2019-10-14 16:47:14*

**Authors:** Mesut Kavak

**Comments:** 3 Pages.

While I was working about some basic physical phenomena, I discovered some geometric relations that also interest mathematics. In this work, I applied the rules I have been proven to P=NP? problem over impossibility of perpendicularity in the universe. It also brings out extremely interesting results out like imaginary numbers which are known as real numbers currently. Also it seems that Euclidean Geometry is impossible. The actual geometry is Riemann Geometry and complex numbers are real.

**Category:** Number Theory

[2134] **viXra:1910.0237 [pdf]**
*submitted on 2019-10-14 22:04:42*

[2133] **viXra:1910.0201 [pdf]**
*submitted on 2019-10-12 14:31:09*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 113 Pages.

In this research thesis, we have described some new mathematical connections between some equations of Dirichlet L-functions, some equations of D-Branes and Rogers-Ramanujan continued fractions.

**Category:** Number Theory

[2132] **viXra:1910.0182 [pdf]**
*submitted on 2019-10-11 22:27:52*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

I was suprised.

**Category:** Number Theory

[2131] **viXra:1910.0180 [pdf]**
*submitted on 2019-10-11 02:45:25*

**Authors:** Pedro Hugo García Peláez

**Comments:** 4 Pages.

Factorization of the numbers of the form n + n ^ 2 it can be done with a certain algorithm.

**Category:** Number Theory

[2130] **viXra:1910.0179 [pdf]**
*submitted on 2019-10-11 02:53:16*

**Authors:** Pedro Hugo García Peláez

**Comments:** 4 Pages.

Los números de la forma n+n^2 se pueden factorizar con un cierto algoritmo.

**Category:** Number Theory

[2129] **viXra:1910.0167 [pdf]**
*submitted on 2019-10-10 16:25:26*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the definition of the Goldbach succession gap (GSG) as not tautologous and contradictory. This means that if the fact of each gap of zero order in a GSG as the difference of squares is based on a contradiction, then Goldbach's strong conjecture and twin primes conjecture are also refuted. Initial proof of the theorem of succession by the inference of induction weakens further the arrival at a definition of GSG . These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[2128] **viXra:1910.0157 [pdf]**
*submitted on 2019-10-10 07:42:33*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 308 Pages.

In this research thesis, we have described some new mathematical connections between some equations of certain Dirichlet series, some equations of D-Branes and Rogers-Ramanujan formulas that link π, e and ϕ.

**Category:** Number Theory

[2127] **viXra:1910.0142 [pdf]**
*submitted on 2019-10-09 07:17:22*

**Authors:** Horacio useche losada

**Comments:** 25 Pages. Primer millón de números primos calculados con una fórmula para el n-ésimo primo

Conseguir una fórmula, un procedimiento o algoritmo para computar el n-
ésimo primo, ha sido siempre un viejo anhelo de los matemáticos. Sin em-
bargo, en la literatura cientı́fica solo se reportan fórmulas basadas en el teo-
rema de Wilson, las cuales, carecen de un valor práctico y solo pueden tener
un interés estrictamente teórico, ya que no se puede llegar muy lejos al in-
tentar su uso en cálculos concretos.
Esta investigación retoma un trabajo del profesor Ramón Fandiño,1 el
cual, presenta en 1980 una relación funcional a partir de la cual se puede com-
putar el n-ésimo primo en función de los n − 1 primos anteriores. Para con-
seguir el objetivo, el profesor Fandiño realiza cinco ajustes, tres por mı́nimos
cuadrados y dos por técnicas implementadas por él mismo, con lo cual con-
sigue calcular los primeros 5000 primos.
Siguiendo la lı́nea de investigación del citado profesor, pero haciendo al-
gunos cambios importantes en el modelo matemático usado y con un menor
número de ajustes, he conseguido computar un millón de números pri-
mos, advirtiendo que es posible computar muchos más,2 si se cuenta con
las herramientas de hardware adecuadas. En esta ocasión, he usado un PC
casero3 , una máquina corriente que logró computar dicha cantidad en tan solo
una hora y 21 minutos! Para hacernos una idea del esfuerzo computacional,
en su momento el profesor Fandiño utilizó, no un PC, sino un computador
de verdad, un IBM 360/44 que era la máquina más poderosa del centro de
cómputo de la UN (y posiblemente de Colombia).4
Con un “juguete”de cómputo, me complace presentar esta cifra que se
enmarca en una polı́tica denominada “resultados sorprendentes con recursos
mediocres”tal y como acontece con otros trabajos de este autor (ver [5], [6], y
[7]). Espero muy pronto superar esta cifra usando un hardware más poderoso,
naturalmente.

**Category:** Number Theory

[2126] **viXra:1910.0137 [pdf]**
*submitted on 2019-10-09 10:09:21*

**Authors:** Miguel Cerdá Bennassar

**Comments:** 35 Pages.

Abstract: I propose a numerical table that demonstrates visually that the sequences formed with Collatz's algorithm always reach 1.

**Category:** Number Theory

[2125] **viXra:1910.0129 [pdf]**
*submitted on 2019-10-09 02:07:26*

[2124] **viXra:1910.0128 [pdf]**
*submitted on 2019-10-08 19:35:03*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate Wüsthofen’s conjecture and counter-example in the title, Benzmüller’s confirmation of Wüsthofen’s conjecture, and Benzmüller’s counter model to Wüsthofen’s counter-example: all four are not tautologous. The claim that the paper in LaTex extension of the proof assistant Isabelle/HOL constitutes a verified proof document is also refuted. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[2123] **viXra:1910.0120 [pdf]**
*submitted on 2019-10-08 00:06:45*

**Authors:** Yuji Masuda

**Comments:** 23 Pages.

This is on primes⑦

**Category:** Number Theory

[2122] **viXra:1910.0117 [pdf]**
*submitted on 2019-10-08 06:37:37*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 153 Pages.

In this research thesis, we have described some new mathematical connections between some equations of certain Dirichlet series, some equations of D-Branes and Rogers-Ramanujan formulas that link π, e and ϕ.

**Category:** Number Theory

[2121] **viXra:1910.0116 [pdf]**
*submitted on 2019-10-08 06:50:03*

**Authors:** Suraj Deshmukh

**Comments:** 7 Pages.

In This paper we will use a simple Logo software to demonstrate a possible
pattern in prime numbers. We Will see how primes show a tendency to retrace the
path of other primes.

**Category:** Number Theory

[2120] **viXra:1910.0115 [pdf]**
*submitted on 2019-10-08 07:07:54*

**Authors:** David Streit, Christoph Benzmüller

**Comments:** 12 Pages.

The present paper is a technical report on 'The Inconsistency of Arithmetic' available on http://vixra.org/abs/1904.0428. It contains a formalized analysis where the authors claim to "constitute a veriﬁed proof document" by an automated verification using the proof assistant 'Isabelle / HOL'. In order to refute the key statement (II) on page 2 of the inconsistency proof, the authors seek to create a countermodel. However, this model is based on an erroneous application of predicate logic. The crucial point is the lemma on page 7 which is proved wrongly. For that statement becoming true, the two sets S1, S2 have to exist for the case that (G) is true and for the case that (G) is false, and not the other way around: if (G) is true there is a pair of unequal sets that does the job and if (G) is false there is another pair.

**Category:** Number Theory

[2119] **viXra:1910.0105 [pdf]**
*submitted on 2019-10-07 08:29:54*

**Authors:** Bassam Abdul-Baki

**Comments:** 31 Pages.

The minimal set for powers of 2 is currently nondeterministic and can be shown to be more complex than previously proposed.

**Category:** Number Theory

[1304] **viXra:2001.0363 [pdf]**
*replaced on 2020-01-19 12:09:04*

**Authors:** Algirdas Antano Maknickas

**Comments:** 1 Page.

These short remarks show deriviation of Birch and Swinnerton-Dyer conjungture. As a consequence new one resulting constant free equality of Birch and Swinnerton-Dyer conjungture proposed

**Category:** Number Theory

[1303] **viXra:2001.0097 [pdf]**
*replaced on 2020-01-08 02:22:57*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 11 Pages. Submitted to the journal Inventiones Matemathicae

In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof that $c<3rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=\frac{3}{e}.e^{ \left(\frac{1}{\epsilon^2} \right)}$ for $0<\epsilon <1$ and $K(\epsilon)=3$ for $\epsilon \geq 1$. Some numerical examples are presented.

**Category:** Number Theory

[1302] **viXra:2001.0084 [pdf]**
*replaced on 2020-01-08 13:30:25*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of a modiﬁed sieve of Eratosthenes to prove the Goldbach conjecture.

**Category:** Number Theory

[1301] **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

[1300] **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

[1299] **viXra:1912.0312 [pdf]**
*replaced on 2019-12-21 17:51:34*

**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

[1298] **viXra:1912.0312 [pdf]**
*replaced on 2019-12-21 09:38:26*

**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

[1297] **viXra:1912.0312 [pdf]**
*replaced on 2019-12-18 08:03:36*

**Authors:** Hassine Saidane

**Comments:** 1 Page.

**Category:** Number Theory

[1296] **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

[1295] **viXra:1912.0245 [pdf]**
*replaced on 2019-12-14 10:25:30*

**Authors:** Edoardo Gueglio

**Comments:** 2 Pages.

CONJECTURES ABOUT THE DIFFERENCE OF THE
SEQUENCE OF RADICALS OF THE NATURAL NUMBERS

**Category:** Number Theory

[1294] **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

[1293] **viXra:1912.0207 [pdf]**
*replaced on 2019-12-13 05:36:02*

**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

[1292] **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

[1291] **viXra:1912.0157 [pdf]**
*replaced on 2019-12-28 17:51:18*

**Authors:** Toshiro Takami

**Comments:** 9 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

[1290] **viXra:1912.0157 [pdf]**
*replaced on 2019-12-08 21:38:41*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

**Category:** Number Theory

[1289] **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). As to other analytic version(s), take note that COS is equivalent to I.

No zeta zero is off the critical line.

Riemann Hypothesis is false if and only if A, B and C are coplanar.

A and B are geometrical objects directly related to (supposed symmetric distinct) zeta roots and C is derived from A and B.

The proof shows that A, B and C are linearly independent, so not coplanar.

**Category:** Number Theory

[1288] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-21 08:48:47*

**Authors:** Ninh Khac Son

**Comments:** 6 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\ Manuscript content: Prove that Goldbach’s conjecture is correct.

**Category:** Number Theory

[1287] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-17 23:06:24*

**Authors:** Ninh Khac Son

**Comments:** 6 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\ Manuscript content: Prove that Goldbach’s conjecture is correct.

**Category:** Number Theory

[1286] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-14 00:42:53*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\ Manuscript content: Prove that Goldbach’s conjecture is correct.

**Category:** Number Theory

[1285] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-13 00:27:25*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\
Manuscript content: Prove that Goldbach’s conjecture is correct

**Category:** Number Theory

[1284] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-10 22:57:48*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\ Manuscript content: Prove that Goldbach’s conjecture is correct

**Category:** Number Theory

[1283] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-09 08:44:29*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: ”Every even integer greater than 2 can be expressed as the sum of two primes”.\\
Manuscript content: Prove that Goldbach’s conjecture is correct

**Category:** Number Theory

[1282] **viXra:1912.0124 [pdf]**
*replaced on 2019-12-08 10:29:51*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.
It states:
*”Every even integer greater than 2 can be expressed as the sum of two primes”.*
**Manuscript content:** Prove that Goldbach’s conjecture is correct.
**Key words:** Prime numbers, Goldbach’s conjecture, number theory.

**Category:** Number Theory

[1281] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-19 16:01:59*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first key step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second key step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered proofs and explanations.

**Category:** Number Theory

[1280] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-16 21:54:51*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first key step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second key step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered proofs and explanations.

**Category:** Number Theory

[1279] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-15 20:00:34*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first key step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second key step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered proofs and explanations.

**Category:** Number Theory

[1278] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-14 05:05:34*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1277] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-12 22:52:47*

**Authors:** John Yuk Ching Ting

**Comments:** 50 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1276] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-12 06:34:16*

**Authors:** John Yuk Ching Ting

**Comments:** 49 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1275] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-11 04:14:28*

**Authors:** John Yuk Ching Ting

**Comments:** 49 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first key step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second key step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered rigorous proofs and explanations.

**Category:** Number Theory

[1274] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-07 20:47:47*

**Authors:** John Yuk Ching Ting

**Comments:** 49 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered rigorous proofs and explanations.

**Category:** Number Theory

[1273] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-05 00:46:19*

**Authors:** John Yuk Ching Ting

**Comments:** 49 Pages. Reverse-engineered Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures. We justify our Reverse-Engineered rigorous proofs and explanations.

**Category:** Number Theory

[1272] **viXra:1911.0201 [pdf]**
*replaced on 2019-11-15 16:55:20*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Triplet and Twin Primes have exactly the same dynamics.
All Prime Triplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Triplet are generated only at (6n -1)(6n+1). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Triplet are 35/12 times of the 3th power distribution of primes, the frequency of occurrence of Prime Triplet is very equal to 0.
However, it is not 0. Therefore, Prime Triplet continue to be generated.
If Prime Triplet is finite, the Primes is finite.
The probability of Prime Triplet 35/12 times of the 3th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Triplet exist forever.

**Category:** Number Theory

[1271] **viXra:1911.0201 [pdf]**
*replaced on 2019-11-15 04:20:33*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Prime Triplet and Twin Primes have exactly the same dynamics.
All Prime Triplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Triplet are generated only at (6n -1)(6n+1). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Triplet are 35/12 times of the 3th power distribution of primes, the frequency of occurrence of Prime Triplet is very equal to 0.
However, it is not 0. Therefore, Prime Triplet continue to be generated.
If Prime Triplet is finite, the Primes is finite.
The probability of Prime Triplet 35/12 times of the 3th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Triplet exist forever.

**Category:** Number Theory

[1270] **viXra:1911.0180 [pdf]**
*replaced on 2019-11-15 06:50:44*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Sextuplet and Twin Primes have exactly the same dynamics.
All Prime Sextuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Sextuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Sextuplet are 48/3 times of the sixth power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Sextuplet continue to be generated.
If Prime Sextuplet is finite, the Primes is finite.
The probability of Prime Sextuplet 48/3 times of the sixth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Sextuplet exist forever.

**Category:** Number Theory

[1269] **viXra:1911.0180 [pdf]**
*replaced on 2019-11-11 16:09:29*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Sextuplet and Twin Primes have exactly the same dynamics.
All Prime Sextuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Sextuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Sextuplet are 48/3 times of the sixth power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Sextuplet continue to be generated.
If Prime Sextuplet is finite, the Primes is finite.
The probability of Prime Sextuplet 48/3 times of the sixth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Sextuplet exist forever.

**Category:** Number Theory

[1268] **viXra:1911.0179 [pdf]**
*replaced on 2019-11-15 18:54:39*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Prime Quintuplet and Twin Primes have exactly the same dynamics.
All Prime Quintuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quintuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quintuplet are 96/3 times of the 5th power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Quintuplet continue to be generated.
If Prime Quintuplet is finite, the Primes is finite.
The probability of Prime Quintuplet 96/3 times of the 5th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quintuplet exist forever.

**Category:** Number Theory

[1267] **viXra:1911.0179 [pdf]**
*replaced on 2019-11-15 04:45:08*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Prime Quintuplet and Twin Primes have exactly the same dynamics.
All Prime Quintuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quintuplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quintuplet are 96/3 times of the 5th power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0.
However, it is not 0. Therefore, Prime Quintuplet continue to be generated.
If Prime Quintuplet is finite, the Primes is finite.
The probability of Prime Quintuplet 96/3 times of the 5th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quintuplet exist forever.

**Category:** Number Theory

[1266] **viXra:1911.0177 [pdf]**
*replaced on 2019-11-15 17:13:24*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Sexy Primes Conjecture were prooved.
Sexy Primes and Twin Primes and Cousin Primes have exactly the same dynamics.
All Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Sexy Primes are generated only at (6n+1)(6n -1). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Sexy Primes are 8/3 times the square of the distribution of primes, the frequency of occurrence of Sexy Primes is very equal to 0.
However, it is not 0. Therefore, Sexy Primes continue to be generated.
If Sexy Primes is finite, the Primes is finite.
Because, Sexy Primes are 8/3 times the square of the distribution of primes.
This is contradictory. Since there are an infinite of Primes.
That is, Sexy Primes exist forever.

**Category:** Number Theory

[1265] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-15 16:58:49*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Prime Quadruplet and Twin Primes have exactly the same dynamics.
All Prime Quadruplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quadruplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quadruplet are 16/3 of the fourth power distribution of primes, the frequency of occurrence of Prime Quadruplet is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated.
If Prime Quadruplet is finite, the Primes is finite.
The probability of Prime Quadruplet 16/3 of the fourth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quadruplet exist forever.

**Category:** Number Theory

[1264] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-12 01:21:56*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

Prime Quadruplet and Twin Primes have exactly the same dynamics.
All Prime Quadruplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Prime Quadruplet are generated only at (6n -1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quadruplet are 16/3 of the fourth power distribution of primes, the frequency of occurrence of Prime Quadruplet is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated.
If Prime Quadruplet is finite, the Primes is finite.
The probability of Prime Quadruplet 16/3 of the fourth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes.
That is, Prime Quadruplet exist forever.

**Category:** Number Theory

[1263] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-08 19:03:42*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

**Category:** Number Theory

[1262] **viXra:1911.0095 [pdf]**
*replaced on 2019-12-04 14:22:27*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the algorithm used for the sieve of Eratosthenes which is not tautologous, hence refuting the conjecture of incompletely predictable problems such as the Riemann zeta function. We also note that the mode of inference used throughout the source is induction. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[1261] **viXra:1911.0093 [pdf]**
*replaced on 2019-12-03 14:31:49*

**Authors:** John Yuk Ching Ting

**Comments:** 48 Pages. Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1260] **viXra:1911.0093 [pdf]**
*replaced on 2019-12-01 20:28:14*

**Authors:** John Yuk Ching Ting

**Comments:** 48 Pages. Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1259] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-29 18:41:54*

**Authors:** John Yuk Ching Ting

**Comments:** 44 Pages. This paper contains Rigorous proofs for open problems Riemann Hypothesis, Polignac's and Twin Prime Conjectures. It also explains the two types of Gram points in Riemann zeta function.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining two types of Gram points), and second step consisting of proving Polignac's and Twin prime conjectures. [Dedication: This landmark research paper is dedicated to my daughter Jelena who was born 13 weeks early on May 14, 2012.]

**Category:** Number Theory

[1258] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-27 23:42:18*

**Authors:** John Yuk Ching Ting

**Comments:** 41 Pages. Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining both Gram points), and second step consisting of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1257] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-26 05:34:11*

**Authors:** John Yuk Ching Ting

**Comments:** 41 Pages. Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function (or proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of applying Information-Complexity conservation to this Sieve result in direct spin-offs from first step consisting of proving Riemann hypothesis (and explaining both Gram points), and second step consisting of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1256] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-18 19:16:28*

**Authors:** John Yuk Ching Ting

**Comments:** 41 Pages. Rigorous proofs for Riemann hypothesis (and explaining the two types of Gram points) and Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Valid and complete mathematical arguments for first key step of converting this function into its continuous format version and second key step of applying Information-Complexity conservation to this Sieve are provided. Direct spin-offs from first step consist of proving Riemann hypothesis (and explaining both types of Gram points) and second step consist of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1255] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-15 16:23:48*

**Authors:** John Yuk Ching Ting

**Comments:** 39 Pages. Rigorous Proofs for Riemann hypothesis (and explaining manifested properties of two types of Gram points), Polignac's and Twin prime conjectures.

Mathematics for Incompletely Predictable Problems is associated with Incompletely Predictable problems containing Incompletely Predictable entities. Nontrivial zeros and two types of Gram points in Riemann zeta function together with prime and composite numbers from Sieve of Eratosthenes are Incompletely Predictable entities. Valid and complete mathematical arguments for first key step of converting this function into its continuous format version and second key step of applying Information-Complexity conservation to this Sieve are provided. Direct spin-offs from first step consist of proving Riemann hypothesis (and explaining both types of Gram points) and second step consist of proving Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1254] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-11 03:44:55*

**Authors:** John Yuk Ching Ting

**Comments:** 39 Pages. Proofs for Riemann hypothesis (and explaining manifested properties of both Gram points), Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1253] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-09 14:28:05*

**Authors:** John Yuk Ching Ting

**Comments:** 39 Pages. Proofs for Riemann hypothesis (and explaining manifested properties of both Gram points), Polignac's and Twin prime conjectures.

**Category:** Number Theory

[1252] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-15 17:22:58*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

Cousin Primes Conjecture were performed using WolframAlpha and Wolfram cloud from the beginning this time, as in the case of the twin primes that we did the other day.
Cousin Primes and Twin Primes have exactly the same dynamics.
All Cousin Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Cousin Primes are generated only at (6n+1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Cousin Primes are 4/3 times the square of the distribution of primes, the frequency of occurrence of Cousin Primes is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated. That is, Cousin Primes exist forever.

**Category:** Number Theory

[1251] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-12 02:39:51*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Cousin Primes Conjecture were performed using WolframAlpha and Wolfram cloud from the beginning this time, as in the case of the twin primes that we did the other day.
Cousin Primes and Twin Primes have exactly the same dynamics.
All Cousin Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number).
In the hexagon, Cousin Primes are generated only at (6n+1)(6n+5). [n is a positive integer]
When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Cousin Primes are 4/3 times the square of the distribution of primes, the frequency of occurrence of Cousin Primes is very equal to 0.
However, it is not 0. Therefore, Cousin Primes continue to be generated. That is, Cousin Primes exist forever.

**Category:** Number Theory

[1250] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-07 04:37:56*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1249] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-06 03:32:05*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1248] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-05 22:10:03*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1247] **viXra:1911.0052 [pdf]**
*replaced on 2019-11-03 01:39:24*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series.

**Category:** Number Theory

[1246] **viXra:1910.0654 [pdf]**
*replaced on 2019-11-02 04:55:39*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Goldbach's conjecture.

**Category:** Number Theory

[1245] **viXra:1910.0650 [pdf]**
*replaced on 2019-11-02 04:39:55*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is proof of a famous formula.

**Category:** Number Theory

[1244] **viXra:1910.0565 [pdf]**
*replaced on 2019-10-28 19:05:17*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration.

**Category:** Number Theory

[1243] **viXra:1910.0551 [pdf]**
*replaced on 2019-10-30 02:26:02*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages. A few corrections and improvements per some suggestions received.

With a strange and ironic twist an open number theory problem, show Zeta(n) is irrational for natural numbers greater than or equal to 2, is solved with the easiest of number theory concepts: the rules of representing fractions with decimals.

**Category:** Number Theory

[1242] **viXra:1910.0499 [pdf]**
*replaced on 2019-10-24 02:25:43*

[1241] **viXra:1910.0349 [pdf]**
*replaced on 2019-11-04 03:39:57*

**Authors:** Y.Mieno

**Comments:** 5 Pages.

A few progressions of the same type and their periodic sequences.

**Category:** Number Theory

[1240] **viXra:1910.0349 [pdf]**
*replaced on 2019-10-20 20:33:11*

**Authors:** Y.Mieno

**Comments:** 5 Pages.

A few progressions of the same type and their periodic sequences.

**Category:** Number Theory

[1239] **viXra:1910.0349 [pdf]**
*replaced on 2019-10-19 21:42:38*

**Authors:** Y.Mieno

**Comments:** 4 Pages.

A few progressions and their periodic sequences.

**Category:** Number Theory

[1238] **viXra:1910.0281 [pdf]**
*replaced on 2019-12-29 20:29:20*

**Authors:** Felix M. Lev

**Comments:** 11 Pages.

The {\it technique} of classical mathematics involves only potential infinity, i.e. infinity is understood only as a limit, and, as a rule, legitimacy of every limit is thoroughly investigated. However, {\it the basis} of classical mathematics does involve actual infinity: the infinite ring of integers $Z$ is the starting point for constructing infinite sets with different cardinalities, and, even in standard textbooks on classical mathematics, it is not even posed a problem whether $Z$ can be treated as a limit of finite sets. On the other hand, finite mathematics starts from the ring $R_p=(0,1,...p-1)$ (where all operations are modulo $p$) and the theory deals only with a finite number of elements. We give a direct proof that $Z$ can be treated as a limit of $R_p$ when $p\to\infty$, and the proof does not involve actual infinity. Then we explain that, as a consequence, finite mathematics is more fundamental than classical one.

**Category:** Number Theory

[1237] **viXra:1910.0105 [pdf]**
*replaced on 2019-10-08 11:41:30*

**Authors:** Bassam Abdul-Baki

**Comments:** 31 Pages.

The minimal set for powers of 2 is currently nondeterministic and can be shown to be more complex than previously proposed.

**Category:** Number Theory