Number Theory

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Recent submissions

Any replacements are listed farther down

[2151] viXra:1910.0261 [pdf] submitted on 2019-10-15 19:05:23

Prime Opinion Part I

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

[2150] viXra:1910.0239 [pdf] submitted on 2019-10-14 16:47:14

Inequality in the Universe, Imaginary Numbers and a Brief Solution to P=NP? Problem

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

[2149] viXra:1910.0237 [pdf] submitted on 2019-10-14 22:04:42

On Prime NumberⅩ

Authors: Yuji Masuda
Comments: 1 Page.

This is on primes⑩
Category: Number Theory

[2148] viXra:1910.0201 [pdf] submitted on 2019-10-12 14:31:09

Further Mathematical Connections Between Some Equations of Dirichlet L-Functions, Some Equations of D-Branes and the Rogers-Ramanujan Continued Fractions. III

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

[2147] viXra:1910.0182 [pdf] submitted on 2019-10-11 22:27:52

On Prime Numbers Ⅸ~Special Edition~

Authors: Yuji Masuda
Comments: 1 Page.

I was suprised.
Category: Number Theory

[2146] viXra:1910.0180 [pdf] submitted on 2019-10-11 02:45:25

Factorization of the Numbers of the Form N + N ^ 2

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

[2145] viXra:1910.0179 [pdf] submitted on 2019-10-11 02:53:16

Factorización de Los Números Naturales de la Forma N+n^2

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

[2144] viXra:1910.0167 [pdf] submitted on 2019-10-10 16:25:26

Refutation of Goldbach Succession Gaps to Prove the Strong Conjecture and Twin Primes

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

[2143] viXra:1910.0157 [pdf] submitted on 2019-10-10 07:42:33

On the Possible Mathematical Connections Between Some Equations of Certain Dirichlet Series, Some Equations of D-Branes and Ramanujan Formula that Link π, e and the Golden Ratio. II

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

[2142] viXra:1910.0142 [pdf] submitted on 2019-10-09 07:17:22

N-Ésimo Primo. Primer Millón de Números Primos Calculados Con Una Fórmula Para el N-Ésimo Primo

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

[2141] viXra:1910.0137 [pdf] submitted on 2019-10-09 10:09:21

The Collatz Conjecture. Order and Harmony in the Sequence Numbers

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

[2140] viXra:1910.0129 [pdf] submitted on 2019-10-09 02:07:26

On Prime NumbersⅧ

Authors: Yuji Masuda
Comments: 49 Pages.

This is primes⑧
Category: Number Theory

[2139] viXra:1910.0128 [pdf] submitted on 2019-10-08 19:35:03

Refutation of Inconsistency of Arithmetic Based on Goldbach Conjecture

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

[2138] viXra:1910.0120 [pdf] submitted on 2019-10-08 00:06:45

On Prime NumbersⅦ

Authors: Yuji Masuda
Comments: 23 Pages.

This is on primes⑦
Category: Number Theory

[2137] viXra:1910.0117 [pdf] submitted on 2019-10-08 06:37:37

On the Possible Mathematical Connections Between Some Equations of Certain Dirichlet Series, Some Equations of D-Branes and Rogers-Ramanujan Formulas that Link π, e and the Golden Ratio. I

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

[2136] viXra:1910.0116 [pdf] submitted on 2019-10-08 06:50:03

Prime Numbers and Its Pattern in Simple Logo

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

[2135] viXra:1910.0115 [pdf] submitted on 2019-10-08 07:07:54

A Technical Report on 'The Inconsistency of Arithmetic'

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

[2134] viXra:1910.0105 [pdf] submitted on 2019-10-07 08:29:54

Minimal Set for Powers of 2

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

[2133] viXra:1910.0081 [pdf] submitted on 2019-10-06 18:13:31

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 19 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[2132] viXra:1910.0077 [pdf] submitted on 2019-10-05 23:10:40

Grimm's Conjecture

Authors: Radomir R.M Majkic
Comments: 3 Pages.

The collection of the consecutive composite integers is the composite connected, and each pair of its distinct integers has a pair of distinct prime divisors. Consequently, it is possible to select a collection of distinct prime divisors, one divisor of one integer of any sequence of the consecutive composed integers.
Category: Number Theory

[2131] viXra:1910.0075 [pdf] submitted on 2019-10-06 03:14:06

(0/0) = 0 = Refuted!

Authors: Ilija Barukčić
Comments: 6 pages. Copyright © 2019 by Ilija Barukčić, Jever, Germany. All rights reserved. Published by:

Objective: The division 0/0 has been investigated by numerous publications while the knowledge that 0/0 = 1 is still not established yet. Methods: A systematic re-analysis of the claim (0/0) = 0 was conducted again. Modus inversus was used to proof the logical consistency of such a claim. Results: The new proof provides strict evidence that 0/0=0 is not correct. Conclusions: 0/0=0 is refuted. Keywords: Division by zero, Modus inversus.
Category: Number Theory

[2130] viXra:1910.0021 [pdf] submitted on 2019-10-01 15:27:10

On the Rogers-Ramanujan Identities and Continued Fractions: New Possible Mathematical Developments and Mathematical Connections with the Mass Value of Candidate “glueball” F0(1710) Meson, Other Particles and the Black Hole Entropies

Authors: Michele Nardelli, Antonio Nardelli
Comments: 210 Pages.

In the present research thesis, we have obtained various and interesting new possible mathematical results concerning the Rogers-Ramanujan identities and some continued fractions. Furthermore, we have described new possible mathematical connections with the mass value of candidate “glueball” f0(1710) meson, other particles and with the Black Hole entropies.
Category: Number Theory

[2129] viXra:1910.0017 [pdf] submitted on 2019-10-01 23:49:24

On Prime Numbers Ⅵ

Authors: Yuji Masuda
Comments: 1 Page.

This is primes⑥.
Category: Number Theory

[2128] viXra:1909.0655 [pdf] submitted on 2019-09-29 17:13:24

Mathematics for Incompletely Predictable Problems Depicting Spin-Offs from Converting Riemann Zeta Function Into Its Continuous Format Version: Paper 1 of 2 Related Papers

Authors: John Ting
Comments: 16 Pages. Proof for Riemann hypothesis and Explanations for Gram points

Mathematics for Incompletely Predictable Problems makes all mathematical arguments valid and complete in [current] Paper 1 (based on first key step of converting Riemann zeta function into its continuous format version) and [next] Paper 2 (based on second key step of applying Information-Complexity conservation to Sieve of Eratosthenes). Nontrivial zeros and two types of Gram points calculated using this function plus prime and composite numbers computed using this Sieve are defined as Incompletely Predictable entities. Euler product formula alternatively and exactly represents Riemann zeta function but utilizes product over prime numbers (instead of summation over natural numbers). Hence prime numbers are encoded in this function demonstrating deep connection between them. Direct spin-offs from first step consist of proving Riemann hypothesis and explaining manifested properties of both Gram points, and from second step consist of proving Polignac's and Twin prime conjectures. These mentioned open problems are defined as Incompletely Predictable problems.
Category: Number Theory

[2127] viXra:1909.0654 [pdf] submitted on 2019-09-29 17:17:10

Mathematics for Incompletely Predictable Problems Depicting Spin-Offs from Applying Information-Complexity Conservation to Sieve of Eratosthenes: Paper 2 of 2 Related Papers

Authors: John Ting
Comments: 15 Pages. Proofs for Polignac's and Twin Prime conjectures

Mathematics for Incompletely Predictable Problems makes all mathematical arguments valid and complete in [previous] Paper 1 (based on first key step of converting Riemann zeta function into its continuous format version) and [current] Paper 2 (based on second key step of applying Information-Complexity conservation to Sieve of Eratosthenes). Nontrivial zeros and two types of Gram points calculated using this function plus prime and composite numbers computed using this Sieve are defined as Incompletely Predictable entities. Euler product formula alternatively and exactly represents Riemann zeta function but utilizes product over prime numbers (instead of summation over natural numbers). Hence prime numbers are encoded in this function demonstrating deep connection between them. Direct spin-offs from first step consist of proving Riemann hypothesis and explaining manifested properties of both Gram points, and from second step consist of proving Polignac's and Twin prime conjectures. These mentioned open problems are defined as Incompletely Predictable problems.
Category: Number Theory

[2126] viXra:1909.0653 [pdf] submitted on 2019-09-29 18:18:41

ζ(4), ζ(6).......ζ(80), ζ(82) Are Irrational Number

Authors: Toshiro Takami
Comments: 22 Pages.

ζ(4), ζ(6).......ζ(80), ζ(82) considered. From these equations, it can be said that ζ(4),ζ(6).......ζ(80),ζ(82) are irrational numbers. ζ(84),ζ(86) etc. can also be expressed by these equations. Because I use π2, these are to be irrational numbers. The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of ζ(2n) is multiplied by π2.
Category: Number Theory

[2125] viXra:1909.0651 [pdf] submitted on 2019-09-29 20:49:56

Refutation of Disproof of One of Cantor's Cardinals

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 two integer lists to map real numbers as Cantor’s cardinals. The disproof of the conjecture that integer infinity is equivalent to real number infinity is not tautologous, so the disproof is refuted. However, this refutation does not automatically confirm the conjecture, forming a non tautologous fragment of the universal logic VŁ4.
Category: Number Theory

[2124] viXra:1909.0649 [pdf] submitted on 2019-09-30 00:52:33

The Yeet Theorem

Authors: Yellocord soc.
Comments: 2 Pages.

Abstract. We provide a surprisingly elementary proof confirming the Yeet Conjecture [Kim14, Yel18], which states that 5^n = n5 for any positive integer n. Moreover, we resolve the ab-Yeet paradox, namely the observation that the quantum state of 5^ab can collapse to either of the values ab or 1. (It has been observed [Lee18] that 5^ab collapses to 1 with probability greater than e for some e > 0.)
Category: Number Theory

[2123] viXra:1909.0618 [pdf] submitted on 2019-09-28 19:28:27

A Disproof to one of Cantor's Cardinals

Authors: Quoss P Wimblik
Comments: 1 Page.

By representing each Integer with 2 Integers we can account for all Real and transcendental numbers given Infinite Intgers.
Category: Number Theory

[2122] viXra:1909.0534 [pdf] submitted on 2019-09-24 07:40:19

The Equation: Psi(q)=2

Authors: Edgar Valdebenito
Comments: 2 Pages.

In this note we give q=0.645..., such that : psi(q)=2, where psi(q) is the Ramanujan's theta function.
Category: Number Theory

[2121] viXra:1909.0532 [pdf] submitted on 2019-09-24 07:43:29

On the Equation: Gamma(x)*gamma(x+1/2)=2

Authors: Edgar Valdebenito
Comments: 3 Pages.

We give the real roots of the equation: gamma(x)*gamma(x+1/2)=2 , x>0 ,where gamma(x) is the Gamma function.
Category: Number Theory

[2120] viXra:1909.0530 [pdf] submitted on 2019-09-24 08:26:16

The Secret of Ishango – On the Helix Structure of Prime Numbers

Authors: Christof Born
Comments: 11 Pages.

The bones of Ishango were found in the 1950s by Belgian archaeologist Jean de Heinzelin near a Palaeolithic residence in Ishango, Africa. Inscriptions, which can be interpreted as numbers, make these bones the oldest mathematical find in human history. There are various scientific papers on the interpretation of the inscriptions. Interestingly, on one of the two bones, we also find the six consecutive prime numbers 5, 7, 11, 13, 17 and 19. Did Stone Age people already know the secret of the prime numbers? This question is explored in my mathematical essay “The Secret of Ishango”: an adventurous journey around the world – from Basel in Switzerland to Erode in India. The presumed connection between the numbers on the bones of Ishango and the structure of the prime numbers is illustrated by a sketch at the end of the text.
Category: Number Theory

[2119] viXra:1909.0515 [pdf] submitted on 2019-09-24 21:25:35

The Requirements on the Non-trivial Roots of the Riemann Zeta via the Dirichlet Eta Sum

Authors: William Blickos
Comments: 11 Pages.

An explanation of the Riemann Hypothesis is given in 8 parts, with the first being a statement of the problem. In the next 3 parts, the complex valued Dirichlet Eta sum, a known equivalence to Riemann Zeta in the critical strip, is split into 8 real valued sums and 2 constants. Part 5 explains a recursive relationship between the 8 sums. Section 6 shows that the sums must individually equal 0. Part 7 details the ratios of the system when all sums equal 0 at once. Finally, part 8 solves the system in terms of the original Dirichlet Eta sum inputs. The result shows that the only possible solution for the real portion of the complex input, commonly labeled a, is that it must equal 1/2, and thus proves Riemann’s suspicion.
Category: Number Theory

[2118] viXra:1909.0504 [pdf] submitted on 2019-09-25 04:22:09

Proof of Goldbach's Conjecture

Authors: Wu Ye TangYin
Comments: 12 Pages. NO

Prime number, compound number, prime factor decomposition, hypothesis. Theme: Integer theory. Push assumptions to infinity according to computational logic Random Extraction Computing Theory Welcome the distinguished gentleman (lady) to comment on my article
Category: Number Theory

[2117] viXra:1909.0495 [pdf] submitted on 2019-09-23 16:00:16

Explicit Upper Bound for all Prime Gaps

Authors: Derek Tucker
Comments: 3 Pages.

Let p_s denote the greatest prime with squared value less than a given number. We call the interval from one prime’s square to the next, a prime’s season. By improving on the well known proof of arbitrarily large prime gaps, here we show that for all seasons, the upper bound of prime gap length is 〖2p〗_s.
Category: Number Theory

[2116] viXra:1909.0473 [pdf] submitted on 2019-09-23 00:57:25

Formula of ζ Even-Numbers

Authors: Toshiro Takami
Comments: 7 Pages.

I published the odd value formula for ζ, but I realized that this was true even when it was even. Therefore, it will be announced.
Category: Number Theory

[2115] viXra:1909.0461 [pdf] submitted on 2019-09-21 12:41:29

A New Primality Test using Fibonnaci Numbers?

Authors: Julia Beauchamp
Comments: 3 Pages.

In this paper, we ask whether a heuristic test for prime numbers can be derived from the Fibonacci numbers. The results below test for values up to $F_{75}$ show that we might have a heuristic test for prime numbers akin to Fermat's Little Theorem.
Category: Number Theory

[2114] viXra:1909.0456 [pdf] submitted on 2019-09-22 02:26:58

On the Ramanujan’s Fundamental Formula for Obtain a Highly Precise Golden Ratio: Mathematical Connections with Black Holes Entropies and Like-Particle Solutions

Authors: Michele Nardelli, Antonio Nardelli
Comments: 79 Pages.

In the present research thesis, we have obtained various and interesting new mathematical connections concerning the fundamental Ramanujan’s formula to obtain a highly precise golden ratio, some sectors of Particle Physics and Black Holes entropies.
Category: Number Theory

[2113] viXra:1909.0385 [pdf] submitted on 2019-09-18 20:47:21

Formula of ζ Odd-Numbers

Authors: Toshiro Takami
Comments: 5 Pages.

I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
Category: Number Theory

[2112] viXra:1909.0384 [pdf] submitted on 2019-09-18 21:28:20

ζ(4), ζ(6).......ζ(108), ζ(110) Are Irrational Number

Authors: Toshiro Takami
Comments: 12 Pages.

ζ(4), ζ(6).......ζ(108), ζ(110) considered. From these equations, it can be said that ζ(4),ζ(6).......ζ(108),ζ(110) are irrational numbers. ζ(112),ζ(114) etc. can also be expressed by these equations. Because I use π2, these are to be irrational numbers. The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of ζ(2n) is multiplied by π2.
Category: Number Theory

[2111] viXra:1909.0378 [pdf] submitted on 2019-09-19 04:18:29

La Conjetura DE Collatz. Orden Y Armonía en Los Números de Las Secuencias.

Authors: Miguel Cerdá Bennassar
Comments: 34 Pages.

Propongo una tabla numérica en la que se demuestra visualmente que las secuencias formadas con el algoritmo de Collatz acaban siempre en el número 1.
Category: Number Theory

[2110] viXra:1909.0370 [pdf] submitted on 2019-09-17 13:19:02

On the Integer Solutions to the Equation X!+x=x^n

Authors: Miika Rankaviita
Comments: 20 Pages. Licencing: CC BY-SA

This thesis explains the solution to the problem of finding all of the integer pair solutions to the equation x!+x=x^n. A detailed explanation is given so that anyone with high school mathematics background can follow the solution. This paper is a translation of my diplom work in Vaasa Lyseo Upper Secondary School.
Category: Number Theory

[2109] viXra:1909.0337 [pdf] submitted on 2019-09-17 00:13:09

A Definiive Proof of the ABC Conjecture

Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages. We give another proof of the conjecture c

In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof the conjecture $c<rad^2(abc)$. 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 $K(\epsilon)=e^{\left(\frac{1}{\epsilon^2} \right)}$. Some numerical examples are presented.
Category: Number Theory

[2108] viXra:1909.0334 [pdf] submitted on 2019-09-17 02:04:19

The Characteristic of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 2 Pages.

In this paper, we propose the axiomatic regularity of prime numbers.
Category: Number Theory

[2107] viXra:1909.0315 [pdf] submitted on 2019-09-15 23:09:11

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 24 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[2106] viXra:1909.0312 [pdf] submitted on 2019-09-14 06:51:50

A Generalization of the Functional Equation of the Riemann zeta Function

Authors: Antoine Balan
Comments: 2 pages, written in french

With help of theta functions, a generalization of the functional equation of the zeta Riemann function can be defined.
Category: Number Theory

[2105] viXra:1909.0305 [pdf] submitted on 2019-09-14 13:53:36

On the Ramanujan Modular Equations, Class Invariants and Mock Theta Functions: New Mathematical Connections with Some Particle-Like Solutions, Black Holes Entropies, ζ(2) and Golden Ratio

Authors: Michele Nardelli, Antonio Nardelli
Comments: 196 Pages.

In the present research thesis, we have obtained various interesting new possible mathematical connections between the Ramanujan Modular Equations, Class Invariants, the Mock Theta Functions, some particle-like solutions, Black Holes entropies, ζ(2) and Golden Ratio
Category: Number Theory

[2104] viXra:1909.0299 [pdf] submitted on 2019-09-15 01:39:20

Collatz Conjecture Explained Through Recursive Functions

Authors: Natalino Sapere
Comments: 9 Pages. None

This paper explains the Collatz Conjecture through the use of recursive functions.
Category: Number Theory

[2103] viXra:1909.0297 [pdf] submitted on 2019-09-15 02:21:15

The Nature of the Φ(m) Function

Authors: Wei Zhang
Comments: 5 Pages.

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, also not clear.
Category: Number Theory

[2102] viXra:1909.0295 [pdf] submitted on 2019-09-15 05:25:01

On Prime NumberⅣ

Authors: Yuji Masuda
Comments: 1 Page.

This is primes④
Category: Number Theory

[2101] viXra:1909.0285 [pdf] submitted on 2019-09-13 19:27:39

Polygonal Numbers in Terms of the Beta Function

Authors: Alfredo Olmos, R. Romyna Olmos
Comments: 7 Pages.

In this article we study some characteristics of polygonal numbers, which are the positive integers that can be ordered, to form a regular polygon. The article is closed, showing the relation of the polygonal numbers, with the Beta function when expressing any polygonal number, as a sum of terms of the Beta function.
Category: Number Theory

[2100] viXra:1909.0178 [pdf] submitted on 2019-09-08 12:33:13

Riemann Hypothesis Proof by Hadamard Product and Monotonicity

Authors: Shekhar Suman
Comments: 5 Pages.

Analytic continuation by hadamard product is strictly monotonic which implies RH
Category: Number Theory

[2099] viXra:1909.0165 [pdf] submitted on 2019-09-09 05:17:04

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 3 page

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[2098] viXra:1909.0154 [pdf] submitted on 2019-09-07 13:41:13

On Prime NumbersⅢ

Authors: Yuji Masuda
Comments: 1 Page.

This is on primes3.
Category: Number Theory

[2097] viXra:1909.0103 [pdf] submitted on 2019-09-05 18:48:43

On Prime Numbers Ⅱ

Authors: Yuji Masuda
Comments: 1 Page.

This is on primes.
Category: Number Theory

[2096] viXra:1909.0059 [pdf] submitted on 2019-09-03 23:11:41

If Riemann’s Zeta Function is True, it Contradicts Zeta’s Dirichlet Series, Causing "Explosion". If it is False, it Causes Unsoundness.

Authors: Ayal Sharon
Comments: 32 Pages. Approx. 7500 words, and approx. 130 references in the bibliography

Riemann's "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane $s \in \mathbb{C}$, $\text{Re}(s)\le1$, (except at $s=1$). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function cannot be both convergent and divergent at any domain value. In other mathematics conjectures and assumed-proven theorems, and in physics, the Riemann Zeta function (or the class of $L$-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle's Laws of Identity, Non-Contradiction, and Excluded Middle. The of Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo-Fraenkel set theory (which was designed to avoid paradoxes). If Riemann's definition of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann's Zeta is false, it renders unsound all theorems and conjectures that falsely assume that it is true. Riemann's Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy's integral theorem.
Category: Number Theory

[2095] viXra:1909.0038 [pdf] submitted on 2019-09-02 12:25:38

Zeroes of The Riemann Zeta Function and Riemann Hypothesis

Authors: Shekhar Suman
Comments: 5 Pages.

Modulus of Hadamard product is shown increasing which proves the Riemann Hypothesis
Category: Number Theory

[2094] viXra:1909.0027 [pdf] submitted on 2019-09-01 12:06:47

Mirror Sieves :Goldbach vs Matiyasevich

Authors: Francis Maleval
Comments: 1 Page.

The sieve of the addition of two prime numbers and the sieve of the product of two natural numbers are linked by a paradox of symmetrical objects. Goldbach's conjecture, additive version of a property of primes, would then have no chance being demonstrated if its multiplicative alter ego remained impenetrable to the disorder of prime numbers.
Category: Number Theory

[2093] viXra:1909.0019 [pdf] submitted on 2019-09-01 21:24:11

Prime Number Pattern 7

Authors: Zeolla Gabriel Martín
Comments: 4 Pages.

This document exposes the construction of infinite patterns for prime numbers smaller than P. In this case, the pattern for prime numbers less than 11 is graphic.
Category: Number Theory

[2092] viXra:1909.0010 [pdf] submitted on 2019-09-01 01:13:44

New Patterns of Modular Arithmetics

Authors: Kurmet Sultan
Comments: 1 Page. This Russian version of the article.

The article reports on the new patterns of modular arithmetic.
Category: Number Theory

[2091] viXra:1908.0617 [pdf] submitted on 2019-08-30 17:11:23

Miroir Aux Alouettes :Goldbach vs Matiyasevich

Authors: Francis Maleval
Comments: 1 Page.

Le crible de l’addition de deux nombres premiers et le crible du produit de deux nombres naturels sont liés par un paradoxe d’objets symétriques. La conjecture de Goldbach, version additive d’une propriété des premiers, n’aurait alors aucune chance d’être un jour démontrée si son alter ego multiplicatif demeurait également impénétrable au désordre, voire au chaos des nombres premiers.
Category: Number Theory

[2090] viXra:1908.0614 [pdf] submitted on 2019-08-31 04:36:55

Division by 0

Authors: Galeotti Giuseppe
Comments: 2 Pages.

the C ensemble is considered close in all the operations but if you divide a number by 0 you will not get a complex number
Category: Number Theory

[2089] viXra:1908.0586 [pdf] submitted on 2019-08-28 08:36:03

Some Fourier Series - Identities

Authors: Edgar Valdebenito
Comments: 3 Pages.

We give some Fourier Series - Identities.
Category: Number Theory

[2088] viXra:1908.0585 [pdf] submitted on 2019-08-28 08:40:22

Numbers: Part 3, " Ramanujan's Integral ".

Authors: Edgar Valdebenito
Comments: 4 Pages.

We recall a Ramanujan's integral: int(f(x),x=0..1)=(pi*pi)/15
Category: Number Theory

[2087] viXra:1908.0568 [pdf] submitted on 2019-08-29 06:21:02

On Prime Numbers

Authors: Yuji Masuda
Comments: 1 Page.

This study focuses on primes.
Category: Number Theory

[2086] viXra:1908.0527 [pdf] submitted on 2019-08-27 04:14:11

Riemann Hypothesis Elementary Proof

Authors: Shekhar Suman
Comments: 4 Pages.

ANALYTIC CONTINUATION AND SIMPLE APPLICATION OF ROLLE'S THEOREM
Category: Number Theory

[2085] viXra:1908.0474 [pdf] submitted on 2019-08-24 02:25:01

Riemann Hypothesis

Authors: Shekhar suman
Comments: 11 Pages. Please send replies at shekharsuman068@gmail.com

Analytic continuation and monotonicity gives the zeroes
Category: Number Theory

[2084] viXra:1908.0427 [pdf] submitted on 2019-08-20 13:26:39

The Riemann Hypothesis Proof

Authors: Shekhar Suman
Comments: 9 Pages. Please read once

We take the integral representation of the Riemann Zeta Function over entire complex plane, except for a pole at 1. Later we draw an equivalent to the Riemann Hypothesis by studying its monotonicity properties.
Category: Number Theory

[2083] viXra:1908.0424 [pdf] submitted on 2019-08-20 15:10:57

The Riemann Hypothesis

Authors: Shekhar Suman
Comments: 7 Pages.

Analytical continuation gives a functional equation having nice properties. Further we give an equivalence of riemann hypotheis through its monotonicity in specific intervals
Category: Number Theory

[2082] viXra:1908.0420 [pdf] submitted on 2019-08-21 05:02:25

A Final Proof of The abc Conjecture

Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages. Comments welcome. Submitted to the Ramanujan Journal.

In this paper, we consider the abc conjecture. As the conjecture c<rad^2(abc) is less open, we give firstly the proof of a modified conjecture that is c<2rad^2(abc). The factor 2 is important for the proof of the new conjecture that represents the key of the proof of the main conjecture. Secondly, the proof of the abc conjecture is given for \epsilon \geq 1, then for \epsilon \in ]0,1[. We choose the constant K(\epsion) as K(\epsilon)=2e^{\frac{1}{\epsilon^2} } for $\epsilon \geq 1 and K(\epsilon)=e^{\frac{1}{\epsilon^2}} for \epsilon \in ]0,1[. Some numerical examples are presented.
Category: Number Theory

[2081] viXra:1908.0416 [pdf] submitted on 2019-08-19 09:50:51

God and Mathematical Beauty IV

Authors: Johannes Abdus Salam
Comments: 1 Page.

I discovered an evidence of the existence of God as the mathematically beautiful equality of the Euler product.
Category: Number Theory

[2080] viXra:1908.0307 [pdf] submitted on 2019-08-14 10:11:12

On Certain Pi_{q}-Identities of W. Gosper

Authors: Bing He
Comments: 16 Pages. All comments are welcome

In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Pi_{q}-identities. As a consequence, a q-identity involving Pi_{q} and Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper.
Category: Number Theory

[2079] viXra:1908.0302 [pdf] submitted on 2019-08-14 14:14:42

The Josephus Numbers

Authors: Kouider Mohammed Ridha
Comments: 3 Pages.

We give explicit formulas to compute the Josephus-numbers where is positive integer . Furthermore we present a new fast algorithm to calculate . We also offer prosperities , and we generalized it for all positive real number non-existent, Finally we give .the proof of properties.
Category: Number Theory

[2078] viXra:1908.0208 [pdf] submitted on 2019-08-11 10:14:19

Rational Distance

Authors: Radomir Majkic
Comments: 3 Pages.

There are countable many rational distance squares, one square for each rational trigonometric Pythagorean pair (s; c) : s^2+c^2=1 and a rational number r:
Category: Number Theory

[2077] viXra:1908.0186 [pdf] submitted on 2019-08-08 23:17:34

The Prime Gaps Between Successive Primes to Ensure that there is Atleast One Prime Between Their Squares Assuming the Truth of the Riemann Hypothesis

Authors: Prashanth R. Rao
Comments: 2 Pages.

Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist between successive primes, so that we can be sure that there is atleast one prime number between their squares.
Category: Number Theory

[2076] viXra:1908.0142 [pdf] submitted on 2019-08-07 08:32:16

Euler Numbers , Catalan's Constant , Number pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

We give some formulas involving Catalan's constant G=0.915965...
Category: Number Theory

[2075] viXra:1908.0140 [pdf] submitted on 2019-08-07 08:41:38

Numbers : Part 1

Authors: Edgar Valdebenito
Comments: 1 Page.

This note presents two Elementary integrals.
Category: Number Theory

[2074] viXra:1908.0139 [pdf] submitted on 2019-08-07 08:44:46

A Note on Ramanujan's Integral

Authors: Edgar Valdebenito
Comments: 2 Pages.

We give some remarks on Ramanujan's integral: int(f(x),x=0..infinite)=(2/3)sqrt(pi).
Category: Number Theory

[2073] viXra:1908.0115 [pdf] submitted on 2019-08-08 03:28:39

The Information Paradox

Authors: Andrea Berdondini
Comments: 4 Pages.

ABSTRACT: The following paradox is based on the consideration that the value of a statistical datum does not represent a useful information, but becomes a useful information only when it is possible to proof that it was not obtained in a random way. In practice, the probability of obtaining the same result randomly must be very low in order to consider the result useful. It follows that the value of a statistical datum is something absolute but its evaluation in order to understand whether it is useful or not is something of relative depending on the actions that have been performed. So two people who analyze the same event, under the same conditions, performing two different procedures obviously find the same value, regarding a statistical parameter, but the evaluation on the importance of the data obtained will be different because it depends on the procedure used. This condition can create a situation like the one described in this paradox, where in one case it is practically certain that the statistical datum is useful, instead in the other case the statistical datum turns out to be completely devoid of value. This paradox wants to bring attention to the importance of the procedure used to extract statistical information; in fact the way in which we act affects the probability of obtaining the same result in a random way and consequently on the evaluation of the statistical parameter.
Category: Number Theory

[2072] viXra:1908.0072 [pdf] submitted on 2019-08-05 02:01:30

Fermat's Last Theorem (Excluding the Case of N=2^t). Unified Method

Authors: Victor Sorokine
Comments: 4 Pages. English version

IN THE FIRST CASE every number (A) is replaced by the sum (A'+A°n) of the last digit and the remainder. After binomial expansion of the Fermat's equality, all the members are combined in two terms: E=A'^n+B'^n-C'^n with the third digit E''', which in one of the n-1 equivalent Fermat's equalities is equal to 2, and the remainder D with the third digit D''', which is equal either to 0, or to n-1, and therefore the third digit of the number A^n+B^n-C^n is different from 0.

IN THE SECOND CASE (for example A=A°n^k, but (BС)'≠0), after having transformed the 3kn-digit ending of the number B into 1 and having left only the last siginificant digits of the numbers A, В, С, simple calculations show that the (3kn-2)-th digit of the number A^n+B^n-C^n is not 0 and does not change after the restoration of all other digits in the numbers A, B, C, because it depends only on the last digit of the number A°.


Category: Number Theory

[2071] viXra:1907.0593 [pdf] submitted on 2019-07-29 06:31:31

Simplified Research for the Constellation of All Roots of Dirichlet Eta Function in Critical Strip

Authors: Leonid Vakhov
Comments: 4 Pages.

The constellation of zeros of Dirichlet eta function is similar to constellation of zeros of important subclass of L-functions (like Dirichlet series etc.). The hereby proposed simplified research can help in researching this important subclass of L-functions.
Category: Number Theory

[2070] viXra:1907.0589 [pdf] submitted on 2019-07-29 09:12:42

Prime Numbers and Twin Prime Numbers Algorithm.

Authors: Zeolla Gabriel Martin
Comments: 24 Pages.

: This article develops an old and well-known expression to obtain prime numbers, composite numbers and twin prime numbers. The conditioning (n) will be the key to make the formula work and the conditioning of the letter (z) will be important for the formula to be efficient.
Category: Number Theory

[2069] viXra:1907.0580 [pdf] submitted on 2019-07-29 22:01:51

An Exact Formula for the Prime Counting Function

Authors: Jose R. Sousa
Comments: 16 Pages. I think this finding may have interesting applications in the study of the Riemann Hypothesis

This article discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of a neat power series for the prime counting function, $\pi(x)$. Among its main findings, we can cite the inversion theorem for Dirichlet series (given $F_a(s)$, we can tell what its associated function, $a(n)$, is), which enabled the creation of a formula for $\pi(x)$ in the first place, and the realization that sums of divisors and the M\"{o}bius function are particular cases of a more general concept. Another conclusion we draw is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.
Category: Number Theory

[2068] viXra:1907.0579 [pdf] submitted on 2019-07-29 22:06:22

On the Limits of a Generalized Harmonic Progression

Authors: Jose R. Sousa
Comments: 7 Pages. Understanding this paper requires a reading of some of the previous papers

This is the fourth paper I'm releasing on the topic of harmonic progressions. Here we address a more complicated problem, namely, the determination of the limiting function of a generalized harmonic progression. It underscores the utility of the formula we derived for $\sum_{j=1}^{n}1/(a\ii j+b)^k$ in $\textit{Complex Harmonic Progression}$ and of results we presented in $\textit{Generalized Harmonic Numbers Revisited}$. Our objective is to create a generating function for $\sum_{k=2}^{\infty}x^k\sum_{j=1}^{\infty}1/(j+b)^k$, with complex $x$ and $b$, whose derivatives at 0 give us the limit of the harmonic progressions (of order 2 and higher) as $n$ approaches infinity.
Category: Number Theory

[2067] viXra:1907.0578 [pdf] submitted on 2019-07-29 22:08:45

Complex Harmonic Progression

Authors: Jose R. Sousa
Comments: 8 Pages. This paper derives a formula that holds for nearly all generalized harmonic progressions

In $\textit{Generalized Harmonic Progression}$, we showed how to create formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, that is, $\sum_{j}1/(a j+b)^k$. Those formulae were more general than the ones we created in $\textit{Generalized Harmonic Numbers Revisited}$. In this new paper we make those formulae even more general by removing the restriction that $a$ and $b$ be integers, in other words, here we address $\sum_{j}1/(a\ii j+b)^k$, where $a$ and $b$ are complex numbers and $\ii$ is the imaginary unity. These new relatively simple formulae always hold, except when $\ii b/a\in \mathbb{Z}$. This paper employs a slightly modified version of the reasoning used previously. Nonetheless, we make another brief exposition of the principle used to derive such formulae.
Category: Number Theory

[2066] viXra:1907.0577 [pdf] submitted on 2019-07-29 22:10:51

Generalized Harmonic Progression

Authors: Jose R. Sousa
Comments: 8 Pages.

This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, more precisely, $\sum_{j=1}^{n}1/(a j+b)^k$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(a,b,n)$ and $S^m_{k}(a,b,n)$ (here, the term $``$harmonic progression$"$ is used loosely, as for some parameter choices, $a$ and $b$, the result may not be a harmonic progression). We provide a generalization of the formulae we created in $\textit{Generalized Harmonic Numbers Revisited}$, which was achieved by using an extension of the reasoning employed before.
Category: Number Theory

[2065] viXra:1907.0558 [pdf] submitted on 2019-07-28 14:39:32

Proof of ∑(n=1,∞)(-1)^n=-1/2

Authors: Yuji Masuda
Comments: 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.
Category: Number Theory

[2064] viXra:1907.0533 [pdf] submitted on 2019-07-26 08:33:19

Hypergeometric Function Identities pi

Authors: Edgar Valdebenito
Comments: 4 Pages.

We give some identities for Pi.
Category: Number Theory

[2063] viXra:1907.0511 [pdf] submitted on 2019-07-27 04:21:14

Infinity and Pythagorean Theory

Authors: Yuji Masuda
Comments: 1 Page.

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.
Category: Number Theory

[2062] viXra:1907.0463 [pdf] submitted on 2019-07-25 00:46:10

Analytic Continuation of the Zeta Function Violates the Law of Non-Contradiction (LNC)

Authors: Ayal Sharon
Comments: Pages.

The Dirichlet series of the Zeta function was long ago proven to be divergent throughout half-plane Re(s) =< 1. If also Riemann's proposition is true, that there exists an "expression" of the Zeta function that is convergent at all values of s (except at s = 1), then the Zeta function is both divergent and convergent throughout half-plane Re(s) =< 1 (except at s = 1). This result violates all three of Aristotle's "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non-Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion": Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann's analytic continuation of the Zeta function violates Cauchy's integral theorem, providing another proof of the invalidity of analytic continuation of the Zeta function. Also, Riemann's Zeta function is one of the L-functions, which are all invalid, because they are generalizations of the invalid analytic continuation of the Zeta function. This result renders unsound all theorems (e.g. Modularity, Fermat's last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann's Zeta function) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar's 3VL, Priest's LP).
Category: Number Theory

[2061] viXra:1907.0437 [pdf] submitted on 2019-07-23 20:48:54

Values of the Riemann Zeta Function by Means of Division by Zero Calculus

Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 12 Pages. In this paper, we will give the values of the Riemann zeta function for any positive integers by means of the division by zero calculus.

In this paper, we will give the values of the Riemann zeta function for any positive integers by means of the division by zero calculus. Zero, division by zero, division by zero calculus, $0/0=1/0=z/0=\tan(\pi/2) = \log 0 =0 $, Laurent expansion, Riemann zeta function, Gamma function, Psi function, Digamma function.
Category: Number Theory

[2060] viXra:1907.0414 [pdf] submitted on 2019-07-23 02:25:12

多与少直接推翻黎曼猜想

Authors: Aaron chau
Comments: 2 Pages.

左边图有二个表示:孪生质数猜想成立。黎曼假设被推翻。右边图表示哥猜是一场没完没了的澄清运动。
Category: Number Theory

[2059] viXra:1907.0400 [pdf] submitted on 2019-07-21 13:24:37

Suggestion to Study Number Theory of Non-Decimal Number Systems

Authors: Jian-ping Gu
Comments: 1 Page.

This paper suggests extending the studies of number theory to non-decimal number systems.
Category: Number Theory

[2058] viXra:1907.0393 [pdf] submitted on 2019-07-21 00:21:22

Equation of Zero

Authors: Yuji Masuda
Comments: 1 Page.

First, ±∞ is constant at any observation point (position).
Category: Number Theory

[2057] viXra:1907.0387 [pdf] submitted on 2019-07-19 07:19:39

Un Millón de Precisiones en Torno a pi

Authors: Horacio useche losada
Comments: 29 Pages. On how to calculate the digits of Pi

El cálculo de los dı́gitos de π ha sido siempre una de las tareas más deseadas por los matemáticos de todos los tiempos, siendo la más antigua de todas. El número π se viene calculando desde la edad de hierro, sin exagerar, y en este documento podrá encontrar un resumen de todos esos esfuerzos con más de 5000 años de historia. Actualmente el record pertenece al fı́sico de partı́culas suizo Peter Trueb, que en noviembre de 2016, encontró 22 459 157 718 361 números decimales de π, completamente verificados. Estos son 2.2 billones de decimales, una can- tidad tan abrumadora que alcanzarı́a para dar 1.2 vueltas al planeta tierra, por el ecuador, y suponiendo cada decimal del tamaño de las letras que ahora lee. Muchos lectores se preguntaran para que sirve calcular tantos dı́gitos de- cimales si para calcular la circuferencia del universo con un error no superior al radio atómico, bastarı́a una precisión de 32 decimales. La respuesta es la misma por la cual el ser humano se empeña en reducir el tiempo de recorrido para los 100 metros planos. Es un sı́mbolo de prepotencia y progreso, del cual, el ser humano, no se puede desprender. Una auténtica demostración de cerebro y máquina que presume del alcance de la especie humana. Para realizar este tipo de esfuerzos, se deben tomar una serie de decisiones concernientes con los algoritmos a usar, esto es, los criterios matemáticos, además de seleccionar las herramientas de software para programar dichos criterios y por último el hardware, o computadores fı́sicos. Todo ello junto, conforma el arsenal de batalla para llevar a cabo hazañas como las de con- quistar nuevos records. Ya se trate de aficionados o matemáticos profesionales, este documento le entrega una revista incremental, desde rústicos y antiguos criterios, hasta los más modernos y sofisticados, usados en la ambiciosa conquista de los dı́gitos de π, que sin duda, le darán lustre a su saber y habilidad. Aquı́, por lo pronto, nos conformamos con llevar al lı́mite de lo posible, las herramientas de hardware casero, con las cuales el lector podrá hacer uso de las mejores teorı́as matemáticas para tener una idea muy fresca y fiel, de las tormentas que se desatan en las cumbres borrascosas de la alta matemática.
Category: Number Theory

[2056] viXra:1907.0378 [pdf] submitted on 2019-07-19 14:15:53

The Goldbach Theorem

Authors: Horacio useche losada
Comments: 33 Pages. The Goldbach's strong conjeture has been proved

Abstract The proof of Goldbach’s strong conjecture is presented, built on the foundations of the theory of gap, which, when combined with certain criteria about the existence of prime numbers in successions, gives us the evidence cited. In reality, We have proof a more general statement in relation to that attributed to Goldbach. As result, it is proved how a even number is the sum of two odd primes, of infinite ways and as a corollary, the conjecture about of the twin primes is also proof.
Category: Number Theory

[2055] viXra:1907.0358 [pdf] submitted on 2019-07-18 16:35:58

Natural Numbers and Their Square Roots Expressed by Constant Phi and 1

Authors: Harry K. Hahn
Comments: 5 pages, 1 drawing

All natural numbers ( 1, 2, 3,…) can be calculated only by using constant Phi (ϕ) and 1. I have found a way to express all natural numbers and their square roots with simple algebraic terms, which are only based on Phi (ϕ) and 1. Further I have found a rule to calculate all natural numbers >10 and their square roots with the help of a general algebraic term. The constant Pi (π) can also be expressed only by using constant Phi and 1 ! It seems that the irrationality of Pi (π) is fundamentally based on the constant Phi and 1, in the same way as the irrationality of all irrational square roots, and all natural numbers seems to be based on constant Phi & 1 ! This is an interesting discovery because it allows to describe many basic geometrical objects like the Platonic Solids only with Phi & 1 ! The result of this discovery may lead to a new base of number theory. Not numbers like 1, 2, 3,… and constants like Pi (π) are the base of number theory ! It seems that only the constant Phi and the base unit 1 ( which shouldn’t be considered as a number ! ) form the base of mathematics and geometry. And constant Phi and the base unit 1 must be considered as the fundamental „space structure constants“ of the real physical world !
Category: Number Theory

[2054] viXra:1907.0357 [pdf] submitted on 2019-07-18 16:41:14

The Ordered Distribution of Natural Numbers on the Square Root Spiral

Authors: Harry K. Hahn
Comments: 35 pages, 17 figures, 3 tables

Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral.
Category: Number Theory

[2053] viXra:1907.0356 [pdf] submitted on 2019-07-18 16:44:28

The Distribution of Prime Numbers on the Square Root Spiral

Authors: Harry K. Hahn
Comments: 44 pages, 26 figures, 7 tables

Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -second difference- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.
Category: Number Theory

[2052] viXra:1907.0355 [pdf] submitted on 2019-07-18 16:47:59

About the Logic of the Prime Number Distribution

Authors: Harry K. Hahn
Comments: 29 pages, 10 figures, 6 tables

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1. All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3. Riemanns Zeta Function also seems to indicate, that there is a logical connection between the mentioned number sequences and the distribution of prime numbers. This connection is indicated by lines in the diagram of the Zeta Function, which are formed by the points s where the Zeta Function is real. Another key role in the distribution of the prime numbers plays the number 5 and its periodic occurrence in the two number sequences SQ1 and SQ2. All non-prime numbers in SQ1 and SQ2 are caused by recurrences of these two number sequences with increasing wave-lengths in themselves, in a similar fashion as Overtones (harmonics) or Undertones derive from a fundamental frequency. On the contrary prime numbers represent spots in these two basic Number Sequences SQ1 and SQ2 where there is no interference caused by these recurring number sequences. The distribution of the non-prime numbers and prime numbers can be described in a graphical way with a -Wave Model- (or Interference Model) -- see Table 2.
Category: Number Theory

[2051] viXra:1907.0354 [pdf] submitted on 2019-07-18 16:53:39

The Distribution of Natural Numbers Divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral

Authors: Harry K. Hahn
Comments: 12 pages, 6 figures

The natural numbers divisible by the Prime Factors 2, 3, 5, 11, 13 and 17 lie on defined spiral graphs, which run through the Square Root Spiral. A mathematical analysis shows, that these spiral graphs are defined by specific quadratic polynomials. Basically all natural number which are divisible by the same prime factor lie on such spiral graphs. And these spiral graphs can be assigned to a certain number of Spiral Graph Systems, which have a defined spatial orientation to each other. This document represents a supplementation to my detailed introduction study to the Square Root Spiral, and it contains the missing diagrams and analyses, showing the distribution of the natural numbers divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral. My introduction study to the Square Root Spiral can also be found in this archive. The title of this study : The ordered distribution of the natural numbers on the Square Root Spiral.
Category: Number Theory

[2050] viXra:1907.0345 [pdf] submitted on 2019-07-17 08:31:03

On: Z*z*z-7*z*z+3*z-1=0

Authors: Edgar Valdebenito
Comments: 3 Pages.

We give a formula for Pi.
Category: Number Theory

[2049] viXra:1907.0303 [pdf] submitted on 2019-07-17 05:02:05

Relative Formula(relationship Between e and π Without i)

Authors: Yuji Masuda
Comments: 1 Page.

This Relative formula shows the relationship between e and π without i.
Category: Number Theory

[2048] viXra:1907.0288 [pdf] submitted on 2019-07-15 08:52:01

Disproof of the Riemann Hypothesis, Long Version

Authors: Igor Hrnčić
Comments: 29 Pages.

In this manuscript we use the Perron formula to connect zeta evaluated on the root free halfplane to zeta evaluated on the critical strip. This is possible since the Perron formula is of the form f(s)=O f(s+w) with O being an integral operator. The variable s+w is on the root free halfplane, and yet s can be on the critical strip. Hence, the Perron formula serves as a form of a functional equation that connects the critical strip with the root free halfplane. Then, one simply notices that in the Perron formula, the left hand side converges only conditionally, whilst the right hand side converges absolutely. This, of course, cannot be, since the left side of an equation is always equal to the right side. This contradiction when examined in detail disproves the Riemann hypothesis. This method is employed on an arbitrary distribution of zeta roots as well, concluding that zeta has a root arbitrarily close to the vertical line passing through unity.
Category: Number Theory

[2047] viXra:1907.0221 [pdf] submitted on 2019-07-13 10:26:58

The Abc Conjecture as Expansion of Powers of Binomials

Authors: Kamal Barghout
Comments: 5 Pages. The manuscript is not to be copied or used in whole or part. The manuscript is copyrighted.

In this note I will show how Beal’s conjecture can be used to study abc conjecture. I will first show how Beal’s conjecture was proved and derive the necessary steps that will lead to further understand the abc conjecture hoping this will aid in proving it. In short, Beal’s conjecture was identified as a univariate Diophantine polynomial identity derived from the binomial identity by expansion of powers of binomials, e.g. the binomial〖 (λx^l+γy^l )〗^n; λ,γ,l,n are positive integers. The idea is that upon expansion and reduction to two terms we can cancel the gcd from the identity equation which leaves the coefficient terms coprime and effectively describes the abc conjecture. To further study the abc terms we need to specifically look for criterion upon which the general property of abc conjecture that states that if the two numbers a and b of the conjecture are divisible by large powers of small primes, a+b tends to be divisible by small powers of large primes which leads to a+b be divisible by large powers of small primes. In this note I only open the door to investigate related possible criterions that may lead to further understand the abc conjecture by expressing it in terms of binomial expansions as Beal’s conjecture was handled.
Category: Number Theory

[2046] viXra:1907.0206 [pdf] submitted on 2019-07-12 23:13:57

On Non-Trivial Zero Point

Authors: Toshiro Takami
Comments: 10 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive. We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5. This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value. We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.
Category: Number Theory

[2045] viXra:1907.0191 [pdf] submitted on 2019-07-12 02:40:19

Algorithms for Testing Prime Factors Against Positive Composite Numbers (Finding the Unique Factorization Domain for Said Composite Numbers) in Base 10: a First Course Into Formal Mathematics

Authors: Labib Zakaria
Comments: 12 Pages. Hopefully this is obvious from the abstract & a quick overview of the paper, but this is not meant to be an immensely technical paper. It is simply meant to be so that people can nurture an appreciation for math. Constructive criticism appreciated.

There exist many algorithms to test the primality of positive natural numbers both proved and unproved, as well as in base 10 and outside base 10. Once the primality of a number has been determined, natural questions are $(1)$ what the unique prime factors of it are and $(2)$ their degree, according to the fundamental theorem of arithmetic. These questions can prove to be useful in beginning to analyze the properties of the number by allowing us to determine the number of (proper) divisors of a number as well as their sum and product. In regards to $(1)$, there are many algorithms that could be applied to determine these prime factors through modular arithmetic algorithms. We will be tackling this question in base 10 specifically by constructing functions as curious mathematicians.
Category: Number Theory

[2044] viXra:1907.0171 [pdf] submitted on 2019-07-11 00:49:20

Proof of Riemann Hypothesis and Other Prime Conjectures

Authors: Surajit Ghosh
Comments: 19 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. Zeta values are renormalised to remove the poles of zeta function. Extending sum to product rule fundamental formula of numbers are defined which then helps proving other prime conjectures namely goldbach conjecture, twin prime conjecture etc.
Category: Number Theory

[2043] viXra:1907.0154 [pdf] submitted on 2019-07-09 18:42:44

A Concise Paper Summarizing a Contradiction with the Riemann Hypothesis

Authors: Viktor Kalaj
Comments: 10 Pages. This paper is rather succinct; it deals with a contradiction while testing the Riemann Zeta function valid on 0 < Re(s) < 1

In this paper, we summarize results of a contradiction while testing the Riemann Hypothesis
Category: Number Theory

Replacements of recent Submissions

[1179] viXra:1910.0105 [pdf] replaced on 2019-10-08 11:41:30

Minimal Set for Powers of 2

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

[1178] viXra:1910.0081 [pdf] replaced on 2019-10-15 23:48:53

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 7 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1177] viXra:1910.0081 [pdf] replaced on 2019-10-13 22:06:58

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 19 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1176] viXra:1910.0081 [pdf] replaced on 2019-10-12 03:17:17

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 33 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1175] viXra:1910.0081 [pdf] replaced on 2019-10-10 23:42:33

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 33 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1174] viXra:1910.0081 [pdf] replaced on 2019-10-10 05:09:13

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 16 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1173] viXra:1910.0081 [pdf] replaced on 2019-10-07 01:35:36

Twin Prime Conjecture (New Edition)

Authors: Toshiro Takami
Comments: 17 Pages.

I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
Category: Number Theory

[1172] viXra:1910.0017 [pdf] replaced on 2019-10-02 05:02:19

On Prime Number Ⅵ2

Authors: Yuji Masuda
Comments: 1 Page.

This is primes⑥.
Category: Number Theory

[1171] viXra:1909.0473 [pdf] replaced on 2019-09-24 21:05:39

Formula of ζ Even-Numbers

Authors: Toshiro Takami
Comments: 16 Pages.

I published the odd value formula for ζ, but I realized that this was true even when it was even. Therefore, it will be announced.
Category: Number Theory

[1170] viXra:1909.0473 [pdf] replaced on 2019-09-24 03:57:49

Formula of ζ Even-Numbers

Authors: Toshiro Takami
Comments: 13 Pages.

I published the odd value formula for ζ, but I realized that this was true even when it was even. Therefore, it will be announced.
Category: Number Theory

[1169] viXra:1909.0461 [pdf] replaced on 2019-10-01 10:30:58

Fibonacci's Answer to Primality Testing?

Authors: Julian TP Beauchamp
Comments: 8 Pages.

In this paper, we consider various approaches to primality testing and then ask whether an effective deterministic test for prime numbers can be found in the Fibonacci numbers.
Category: Number Theory

[1168] viXra:1909.0461 [pdf] replaced on 2019-09-28 08:59:54

Fibonacci's Answer to Primality Testing?

Authors: Julian TP Beauchamp
Comments: 7 Pages.

In this paper, we consider various approaches to primality testing and then ask whether an effective deterministic test for prime numbers can be found in the Fibonacci numbers.
Category: Number Theory

[1167] viXra:1909.0385 [pdf] replaced on 2019-09-29 23:13:57

Formula of ζ Odd-Numbers

Authors: Toshiro Takami
Comments: 33 Pages.

I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
Category: Number Theory

[1166] viXra:1909.0385 [pdf] replaced on 2019-09-28 18:38:00

Formula of ζ Odd-Numbers

Authors: Toshiro Takami
Comments: 8 Pages.

I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
Category: Number Theory

[1165] viXra:1909.0385 [pdf] replaced on 2019-09-24 18:52:15

Formula of ζ Odd-Numbers

Authors: Toshiro Takami
Comments: 6 Pages.

I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
Category: Number Theory

[1164] viXra:1909.0385 [pdf] replaced on 2019-09-22 00:51:56

Formula of ζ Odd-Numbers

Authors: Toshiro Takami
Comments: 11 Pages.

I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
Category: Number Theory

[1163] viXra:1909.0384 [pdf] replaced on 2019-09-23 03:33:35

ζ(4), ζ(6).......ζ(108), ζ(110) Are Irrational Number

Authors: Toshiro Takami
Comments: 9 Pages.

ζ(4), ζ(6).......ζ(108), ζ(110) considered. From these equations, it can be said that ζ(4),ζ(6).......ζ(108),ζ(110) are irrational numbers. ζ(112),ζ(114) etc. can also be expressed by these equations. Because I use π2, these are to be irrational numbers. The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of ζ(2n) is multiplied by π2.
Category: Number Theory

[1162] viXra:1909.0315 [pdf] replaced on 2019-09-27 19:16:23

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 34 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[1161] viXra:1909.0315 [pdf] replaced on 2019-09-25 00:01:29

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 32 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[1160] viXra:1909.0315 [pdf] replaced on 2019-09-20 03:32:04

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 37 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[1159] viXra:1909.0315 [pdf] replaced on 2019-09-19 03:16:57

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 35 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[1158] viXra:1909.0315 [pdf] replaced on 2019-09-17 08:58:39

ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number

Authors: Toshiro Takami
Comments: 38 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
Category: Number Theory

[1157] viXra:1909.0165 [pdf] replaced on 2019-10-13 18:05:39

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 7 pages

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[1156] viXra:1909.0165 [pdf] replaced on 2019-10-05 12:02:53

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 6 pages

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[1155] viXra:1909.0165 [pdf] replaced on 2019-09-30 03:17:28

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 5 pages

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[1154] viXra:1909.0165 [pdf] replaced on 2019-09-28 17:23:57

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 4 pages

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[1153] viXra:1909.0165 [pdf] replaced on 2019-09-11 01:57:47

Proof of Goldbach's Strong Conjecture

Authors: Sitangsu Maitra
Comments: 4 pages

Proof of Goldbach's strong conjecture in a different way
Category: Number Theory

[1152] viXra:1908.0302 [pdf] replaced on 2019-08-29 03:23:02

The Josephus Numbers

Authors: Kouider Mohammed Ridha
Comments: 3 Pages.

According to Josephuse history we present a new numbers called The josephuse numbers. Hence we give explicit formulas to compute the Josephus-numbers J(n)where n is positive integer . Furthermore we present a new fast algorithm to calculate J(n). We also offer prosperities , and we generalized it for all positive real number non-existent, Finally we give .the proof of properties.
Category: Number Theory

[1151] viXra:1908.0186 [pdf] replaced on 2019-08-13 16:53:26

The Prime Gaps Between Successive Primes to Ensure that there is Atleast One Prime Between Their Squares Assuming the Truth of the Riemann Hypothesis

Authors: Prashanth R. Rao
Comments: 2 Pages.

Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist between successive primes, so that we can be sure that there is atleast one prime number between their squares.
Category: Number Theory

[1150] viXra:1907.0558 [pdf] replaced on 2019-07-29 10:29:00

Proof of ∑(n=1,∞)(-1)^n=-1/2

Authors: Yuji Masuda
Comments: 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.
Category: Number Theory

[1149] viXra:1907.0558 [pdf] replaced on 2019-07-28 15:32:20

Proof of ∑(n=1,∞)(-1)^n=-1/2

Authors: Yuji Masuda
Comments: 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.
Category: Number Theory

[1148] viXra:1907.0521 [pdf] replaced on 2019-07-26 22:32:15

Infinity and Pythagorean Theorem

Authors: Yuji Masuda
Comments: 1 Page.

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.
Category: Number Theory

[1147] viXra:1907.0378 [pdf] replaced on 2019-09-22 12:08:14

The Goldbach Theorem

Authors: Horacio Useche Losada
Comments: 33 Pages.

The proof of Goldbach’s strong conjecture is presented, built on the foundations of the theory of gap, which, when combined with certain criteria about the existence of prime numbers in successions, gives us the evidence cited. In reality, We have proof a more general statement in relation to that attributed to Goldbach. As result, it is proved how a even number is the sum of two odd primes, of infinite ways and as a corollary, the conjecture about of the twin primes is also proof.
Category: Number Theory

[1146] viXra:1907.0303 [pdf] replaced on 2019-07-19 23:41:18

Relative Formula (Relationship Between e and π Without i)

Authors: Yuji Masuda
Comments: 1 Page.

This relative formula shows The relationship between napier number e and π without imaginary unit i.
Category: Number Theory

[1145] viXra:1907.0206 [pdf] replaced on 2019-07-25 07:13:55

On Non-Trivial Zero Point

Authors: Toshiro Takami
Comments: 39 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive. We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5. This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value. We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.
Category: Number Theory

[1144] viXra:1907.0206 [pdf] replaced on 2019-07-24 01:12:26

On Non-Trivial Zero Point

Authors: Toshiro Takami
Comments: 24 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive. We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5. This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value. We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.
Category: Number Theory