**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(2) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(7) - 1203(6) - 1204(6) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(13) - 1406(18) - 1407(33) - 1408(50) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(17) - 1506(12) - 1507(15) - 1508(14) - 1509(13) - 1510(11) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(53) - 1605(28) - 1606(17) - 1607(17) - 1608(15) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(23) - 1703(25) - 1704(32) - 1705(25) - 1706(25) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(23) - 1712(34)

2018 - 1801(31) - 1802(20) - 1803(22) - 1804(25) - 1805(31) - 1806(16) - 1807(18) - 1808(14) - 1809(22) - 1810(16) - 1811(25) - 1812(29)

2019 - 1901(12) - 1902(11) - 1903(21) - 1904(25) - 1905(23) - 1906(42) - 1907(42) - 1908(20) - 1909(37) - 1910(23)

Any replacements are listed farther down

[2151] **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

[2150] **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

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

[2148] **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

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

[2139] **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

[2138] **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

[2137] **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

[2136] **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

[2135] **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

[2134] **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

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

**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*

**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*

**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*

**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*

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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°.

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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

[1179] **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

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

**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*

**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*

**Authors:** Toshiro Takami

**Comments:** 33 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 33 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 16 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 17 Pages.

**Category:** Number Theory

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

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

**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*

**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*

**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*

**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*

**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*

**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*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

**Category:** Number Theory

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

**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*

**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*

**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*

**Authors:** Toshiro Takami

**Comments:** 37 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 35 Pages.

**Category:** Number Theory

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

**Authors:** Toshiro Takami

**Comments:** 38 Pages.

**Category:** Number Theory

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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