**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(3) - 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(18) - 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(19) - 1803(22) - 1804(25) - 1805(31) - 1806(16) - 1807(18) - 1808(13) - 1809(22) - 1810(16) - 1811(24) - 1812(29)

2019 - 1901(11) - 1902(11) - 1903(21) - 1904(25) - 1905(22) - 1906(42) - 1907(42) - 1908(20) - 1909(35) - 1910(49) - 1911(29) - 1912(8)

Any replacements are listed farther down

[2206] **viXra:1912.0124 [pdf]**
*submitted on 2019-12-06 14:34:36*

**Authors:** Ninh Khac Son

**Comments:** 5 Pages.

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

**Category:** Number Theory

[2205] **viXra:1912.0119 [pdf]**
*submitted on 2019-12-06 02:07:40*

**Authors:** Takamasa Noguchi

**Comments:** 4 Pages. Explanation of effective prime number judgment method even for Carmichael number.

Explanation of effective prime number judgment method even for Carmichael number.
This method of judgment does not give a 100% correct answer.
Care must be taken especially for (n=p^k (P=Prime)) with primitive roots.

**Category:** Number Theory

[2204] **viXra:1912.0117 [pdf]**
*submitted on 2019-12-06 04:20:47*

**Authors:** Matanari Shimoinuda

**Comments:** 32 Pages.

In this article, we review the local Langlands program. I hope that this article is a useful guide to understand Langlands program.

**Category:** Number Theory

[2203] **viXra:1912.0115 [pdf]**
*submitted on 2019-12-06 04:57:22*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

I proved the Twin Prime Conjecture by using Clement's theorem.\\
I was able to transform below.\ (n\ is\ positive\ integer)\\
\\
4$\times(6n-2)!+6n+3\equiv0\ \ \ ( mod\ \ (6n-1)(6n+1)).$ \\
Even if the number(n) reaches the limit, use {n=x+1} from {n=x}.\ \ Larger Twin Primes are born.\ \ Repeat this.\\
That is, Twin Primes exist forever.\\

**Category:** Number Theory

[2202] **viXra:1912.0108 [pdf]**
*submitted on 2019-12-06 06:49:41*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 83 Pages.

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

**Category:** Number Theory

[2201] **viXra:1912.0087 [pdf]**
*submitted on 2019-12-05 05:25:17*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 133 Pages.

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

**Category:** Number Theory

[2200] **viXra:1912.0082 [pdf]**
*submitted on 2019-12-03 20:13:14*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[2199] **viXra:1912.0028 [pdf]**
*submitted on 2019-12-02 13:53:35*

**Authors:** BOUARFAOUI Saïd

**Comments:** 8 Pages. Some studies on the 3n+1 problem

In the present paper, we study the 3n+1 problem to know if there is a theoretical number such as the number never land on the cycle {4;2;1}and grows continually.We will specify the general formula such as the number of uneven steps (3��+1)and even steps (��/2)are equals.We want to know also the uneven number form 2^��+2^0 and its composition during ��uneven (3��+1)and even steps (��/2).The composition number 2��+20will be viewed in two ways :1. When the powers which compose the numbers are expanded. We will recognize the Pascal’s triangle.2. When the powers which compose the numbers are agglutinate. We will recognize (3^��) in base 2.

**Category:** Number Theory

[2198] **viXra:1911.0525 [pdf]**
*submitted on 2019-11-30 10:17:22*

**Authors:** Miguel Cerdá Bennassar

**Comments:** 7 Pages.

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

**Category:** Number Theory

[2197] **viXra:1911.0484 [pdf]**
*submitted on 2019-11-29 09:47:14*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

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

**Category:** Number Theory

[2196] **viXra:1911.0477 [pdf]**
*submitted on 2019-11-28 02:22:12*

**Authors:** Dmitri Martila

**Comments:** 3 Pages.

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

**Category:** Number Theory

[2195] **viXra:1911.0426 [pdf]**
*submitted on 2019-11-25 06:34:15*

**Authors:** Juan Moreno Borrallo

**Comments:** 4 Pages.

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

**Category:** Number Theory

[2194] **viXra:1911.0316 [pdf]**
*submitted on 2019-11-18 11:12:58*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

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

**Category:** Number Theory

[2193] **viXra:1911.0310 [pdf]**
*submitted on 2019-11-18 05:46:30*

**Authors:** Aaron chau

**Comments:** 2 Pages.

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

**Category:** Number Theory

[2192] **viXra:1911.0308 [pdf]**
*submitted on 2019-11-18 07:05:36*

**Authors:** James Edwin Rock

**Comments:** 2 Pages.

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

**Category:** Number Theory

[2191] **viXra:1911.0287 [pdf]**
*submitted on 2019-11-17 08:18:55*

**Authors:** Richard J. Mathar

**Comments:** 39 Pages.

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

**Category:** Number Theory

[2190] **viXra:1911.0236 [pdf]**
*submitted on 2019-11-13 14:02:48*

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

**Comments:** 35 Pages.

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

**Category:** Number Theory

[2189] **viXra:1911.0208 [pdf]**
*submitted on 2019-11-11 14:09:39*

**Authors:** Colin James III

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

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

**Category:** Number Theory

[2188] **viXra:1911.0201 [pdf]**
*submitted on 2019-11-11 02:24:22*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[2187] **viXra:1911.0198 [pdf]**
*submitted on 2019-11-11 07:34:34*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 172 Pages.

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

**Category:** Number Theory

[2186] **viXra:1911.0180 [pdf]**
*submitted on 2019-11-09 16:17:51*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[2185] **viXra:1911.0179 [pdf]**
*submitted on 2019-11-09 17:24:44*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[2184] **viXra:1911.0177 [pdf]**
*submitted on 2019-11-09 01:24:48*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

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

**Category:** Number Theory

[2183] **viXra:1911.0144 [pdf]**
*submitted on 2019-11-08 07:33:56*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[2182] **viXra:1911.0116 [pdf]**
*submitted on 2019-11-06 12:19:15*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 117 Pages.

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

**Category:** Number Theory

[2181] **viXra:1911.0111 [pdf]**
*submitted on 2019-11-06 01:41:19*

**Authors:** Wei Zhang

**Comments:** 6 Pages.

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

**Category:** Number Theory

[2180] **viXra:1911.0095 [pdf]**
*submitted on 2019-11-06 10:12:07*

**Authors:** Colin James III

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

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

**Category:** Number Theory

[2179] **viXra:1911.0093 [pdf]**
*submitted on 2019-11-05 14:55:47*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[2178] **viXra:1911.0083 [pdf]**
*submitted on 2019-11-05 04:38:15*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[2177] **viXra:1911.0079 [pdf]**
*submitted on 2019-11-05 09:08:26*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 91 Pages.

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

**Category:** Number Theory

[2176] **viXra:1911.0061 [pdf]**
*submitted on 2019-11-04 01:47:31*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series part3.

**Category:** Number Theory

[2175] **viXra:1911.0052 [pdf]**
*submitted on 2019-11-03 01:40:08*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series.

**Category:** Number Theory

[2174] **viXra:1911.0049 [pdf]**
*submitted on 2019-11-03 02:40:36*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series part2.

**Category:** Number Theory

[2173] **viXra:1911.0044 [pdf]**
*submitted on 2019-11-03 07:05:12*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

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

**Category:** Number Theory

[2172] **viXra:1911.0020 [pdf]**
*submitted on 2019-11-01 16:12:52*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 35 Pages.

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

**Category:** Number Theory

[2171] **viXra:1911.0019 [pdf]**
*submitted on 2019-11-01 17:30:43*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Riemann hypothesis.

**Category:** Number Theory

[2170] **viXra:1911.0002 [pdf]**
*submitted on 2019-11-01 09:10:26*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

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

**Category:** Number Theory

[2169] **viXra:1910.0654 [pdf]**
*submitted on 2019-10-31 21:02:38*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Goldbach's conjecture.

**Category:** Number Theory

[2168] **viXra:1910.0650 [pdf]**
*submitted on 2019-10-31 03:23:41*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is proof of a famous formula.

**Category:** Number Theory

[2167] **viXra:1910.0636 [pdf]**
*submitted on 2019-10-30 22:53:32*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration3.

**Category:** Number Theory

[2166] **viXra:1910.0635 [pdf]**
*submitted on 2019-10-29 22:55:16*

**Authors:** Kouji Takaki

**Comments:** 14 Pages.

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

**Category:** Number Theory

[2165] **viXra:1910.0634 [pdf]**
*submitted on 2019-10-30 00:30:29*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration２.

**Category:** Number Theory

[2164] **viXra:1910.0565 [pdf]**
*submitted on 2019-10-27 23:13:37*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

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

**Category:** Number Theory

[2163] **viXra:1910.0563 [pdf]**
*submitted on 2019-10-27 02:56:14*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

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

**Category:** Number Theory

[2162] **viXra:1910.0558 [pdf]**
*submitted on 2019-10-27 07:27:30*

**Authors:** William F. Gilreath

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

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

**Category:** Number Theory

[2161] **viXra:1910.0551 [pdf]**
*submitted on 2019-10-27 12:15:29*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

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

**Category:** Number Theory

[2160] **viXra:1910.0549 [pdf]**
*submitted on 2019-10-26 17:08:03*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 108 Pages.

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

**Category:** Number Theory

[2159] **viXra:1910.0538 [pdf]**
*submitted on 2019-10-26 07:33:48*

[2158] **viXra:1910.0499 [pdf]**
*submitted on 2019-10-24 02:15:04*

[2157] **viXra:1910.0494 [pdf]**
*submitted on 2019-10-24 05:10:18*

**Authors:** Zhiping Dai

**Comments:** 7 Pages.

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

**Category:** Number Theory

[2156] **viXra:1910.0475 [pdf]**
*submitted on 2019-10-23 21:44:14*

**Authors:** Derek Tucker

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

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

**Category:** Number Theory

[2155] **viXra:1910.0444 [pdf]**
*submitted on 2019-10-23 12:21:40*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 152 Pages.

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

**Category:** Number Theory

[2154] **viXra:1910.0411 [pdf]**
*submitted on 2019-10-21 03:47:35*

[2153] **viXra:1910.0395 [pdf]**
*submitted on 2019-10-21 11:38:49*

**Authors:** Siddharth Bhatt

**Comments:** 1 Page.

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

**Category:** Number Theory

[2152] **viXra:1910.0367 [pdf]**
*submitted on 2019-10-20 11:45:58*

**Authors:** Derek Tucker

**Comments:** 3 Pages.

Proof of Legendre's conjecture by elementary means.

**Category:** Number Theory

[2151] **viXra:1910.0366 [pdf]**
*submitted on 2019-10-20 11:58:01*

**Authors:** Abdelmajid Ben Hadj Salem

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

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

**Category:** Number Theory

[2150] **viXra:1910.0365 [pdf]**
*submitted on 2019-10-20 11:57:05*

**Authors:** Derek Tucker

**Comments:** 1 Page.

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

**Category:** Number Theory

[2149] **viXra:1910.0364 [pdf]**
*submitted on 2019-10-19 15:17:19*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 59 Pages.

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

**Category:** Number Theory

[2148] **viXra:1910.0349 [pdf]**
*submitted on 2019-10-19 07:42:17*

**Authors:** Y.Mieno

**Comments:** 3 Pages.

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

**Category:** Number Theory

[2147] **viXra:1910.0322 [pdf]**
*submitted on 2019-10-18 12:33:20*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 201 Pages.

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

**Category:** Number Theory

[2146] **viXra:1910.0317 [pdf]**
*submitted on 2019-10-17 02:18:31*

**Authors:** Wu Ye TangYin

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

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

**Category:** Number Theory

[2145] **viXra:1910.0316 [pdf]**
*submitted on 2019-10-17 02:46:52*

**Authors:** Wu Ye TangYin

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

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

**Category:** Number Theory

[2144] **viXra:1910.0281 [pdf]**
*submitted on 2019-10-16 00:11:18*

**Authors:** Felix M. Lev

**Comments:** 9 Pages.

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

**Category:** Number Theory

[2143] **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

[2142] **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

[2141] **viXra:1910.0237 [pdf]**
*submitted on 2019-10-14 22:04:42*

[2140] **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

[2139] **viXra:1910.0182 [pdf]**
*submitted on 2019-10-11 22:27:52*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

I was suprised.

**Category:** Number Theory

[2138] **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

[2137] **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

[2136] **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

[2135] **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

[2134] **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

[2133] **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

[2132] **viXra:1910.0129 [pdf]**
*submitted on 2019-10-09 02:07:26*

[2131] **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

[2130] **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

[2129] **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

[2128] **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

[2127] **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

[2126] **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

[2125] **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

[2124] **viXra:1910.0077 [pdf]**
*submitted on 2019-10-05 23:10:40*

**Authors:** Radomir Majkic

**Comments:** 3 Pages. The author's name typing error s corrected.

The collection of the consecutive composite integers is the composite connected, and no pair of its distinct integers may be generated by a single prime number.
Consequently, it is possible to select at least one collection of distinct prime divisors by extracting only one prime from every single integer of the collection, and the Grimm's Conjecture holds.

**Category:** Number Theory

[2123] **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

[2122] **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

[2121] **viXra:1910.0017 [pdf]**
*submitted on 2019-10-01 23:49:24*

[2120] **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

[2119] **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

[2118] **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

[2117] **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

[2116] **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

[2115] **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

[2114] **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

[2113] **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

[2112] **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

[2111] **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

[2110] **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

[2109] **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

[2108] **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

[2107] **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

[2106] **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

[2105] **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

[2104] **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

[2103] **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

[2102] **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

[2101] **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

[2100] **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

[2099] **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

[2098] **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

[2097] **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

[2096] **viXra:1909.0295 [pdf]**
*submitted on 2019-09-15 05:25:01*

[2095] **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

[2094] **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

[2093] **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

[2092] **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

[2091] **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

[2090] **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

[2089] **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

[2088] **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

[2087] **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

[2086] **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

[1265] **viXra:1912.0082 [pdf]**
*replaced on 2019-12-05 00:46:19*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1264] **viXra:1911.0201 [pdf]**
*replaced on 2019-11-15 16:55:20*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1263] **viXra:1911.0201 [pdf]**
*replaced on 2019-11-15 04:20:33*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1262] **viXra:1911.0180 [pdf]**
*replaced on 2019-11-15 06:50:44*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1261] **viXra:1911.0180 [pdf]**
*replaced on 2019-11-11 16:09:29*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1260] **viXra:1911.0179 [pdf]**
*replaced on 2019-11-15 18:54:39*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1259] **viXra:1911.0179 [pdf]**
*replaced on 2019-11-15 04:45:08*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1258] **viXra:1911.0177 [pdf]**
*replaced on 2019-11-15 17:13:24*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1257] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-15 16:58:49*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1256] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-12 01:21:56*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

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

**Category:** Number Theory

[1255] **viXra:1911.0144 [pdf]**
*replaced on 2019-11-08 19:03:42*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

**Category:** Number Theory

[1254] **viXra:1911.0095 [pdf]**
*replaced on 2019-12-04 14:22:27*

**Authors:** Colin James III

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

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

**Category:** Number Theory

[1253] **viXra:1911.0093 [pdf]**
*replaced on 2019-12-03 14:31:49*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1252] **viXra:1911.0093 [pdf]**
*replaced on 2019-12-01 20:28:14*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1251] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-29 18:41:54*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1250] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-27 23:42:18*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1249] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-26 05:34:11*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1248] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-18 19:16:28*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1247] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-15 16:23:48*

**Authors:** John Yuk Ching Ting

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

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

**Category:** Number Theory

[1246] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-11 03:44:55*

**Authors:** John Yuk Ching Ting

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

**Category:** Number Theory

[1245] **viXra:1911.0093 [pdf]**
*replaced on 2019-11-09 14:28:05*

**Authors:** John Yuk Ching Ting

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

**Category:** Number Theory

[1244] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-15 17:22:58*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

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

**Category:** Number Theory

[1243] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-12 02:39:51*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1242] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-07 04:37:56*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1241] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-06 03:32:05*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1240] **viXra:1911.0083 [pdf]**
*replaced on 2019-11-05 22:10:03*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1239] **viXra:1911.0052 [pdf]**
*replaced on 2019-11-03 01:39:24*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of some convergent series.

**Category:** Number Theory

[1238] **viXra:1910.0654 [pdf]**
*replaced on 2019-11-02 04:55:39*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of Goldbach's conjecture.

**Category:** Number Theory

[1237] **viXra:1910.0650 [pdf]**
*replaced on 2019-11-02 04:39:55*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is proof of a famous formula.

**Category:** Number Theory

[1236] **viXra:1910.0635 [pdf]**
*replaced on 2019-12-06 19:52:23*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1235] **viXra:1910.0635 [pdf]**
*replaced on 2019-12-05 17:33:03*

**Authors:** Kouji Takaki

**Comments:** 14 Pages.

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

**Category:** Number Theory

[1234] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-27 08:18:00*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1233] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-23 14:53:59*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1232] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-23 06:35:44*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1231] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-22 01:33:49*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1230] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-16 09:07:05*

**Authors:** Kouji Takaki

**Comments:** 13 Pages.

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

**Category:** Number Theory

[1229] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-15 02:13:18*

**Authors:** Kouji Takaki

**Comments:** 15 Pages.

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

**Category:** Number Theory

[1228] **viXra:1910.0635 [pdf]**
*replaced on 2019-11-14 03:51:44*

**Authors:** Kouji Takaki

**Comments:** 15 Pages.

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

**Category:** Number Theory

[1227] **viXra:1910.0565 [pdf]**
*replaced on 2019-10-28 19:05:17*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is collaboration.

**Category:** Number Theory

[1226] **viXra:1910.0551 [pdf]**
*replaced on 2019-10-30 02:26:02*

**Authors:** Timothy W. Jones

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

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

**Category:** Number Theory

[1225] **viXra:1910.0499 [pdf]**
*replaced on 2019-10-24 02:25:43*

[1224] **viXra:1910.0349 [pdf]**
*replaced on 2019-11-04 03:39:57*

**Authors:** Y.Mieno

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1223] **viXra:1910.0349 [pdf]**
*replaced on 2019-10-20 20:33:11*

**Authors:** Y.Mieno

**Comments:** 5 Pages.

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

**Category:** Number Theory

[1222] **viXra:1910.0349 [pdf]**
*replaced on 2019-10-19 21:42:38*

**Authors:** Y.Mieno

**Comments:** 4 Pages.

A few progressions and their periodic sequences.

**Category:** Number Theory

[1221] **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

[1220] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-26 18:47:27*

**Authors:** Toshiro Takami

**Comments:** 10 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

[1219] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-25 13:10:45*

**Authors:** Toshiro Takami

**Comments:** 11 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

[1218] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-15 03:19:02*

**Authors:** Toshiro Takami

**Comments:** 13 Pages.

**Category:** Number Theory

[1217] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-14 16:54:23*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1216] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-13 23:35:12*

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

**Category:** Number Theory

[1215] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-12 20:33:54*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1214] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-12 02:53:04*

**Authors:** Toshiro Takami

**Comments:** 13 Pages.

**Category:** Number Theory

[1213] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-07 04:36:34*

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

**Category:** Number Theory

[1212] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-06 03:30:57*

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

**Category:** Number Theory

[1211] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-05 22:08:39*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

**Category:** Number Theory

[1210] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-04 23:37:47*

**Authors:** Toshiro Takami

**Comments:** 13 Pages.

**Category:** Number Theory

[1209] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-04 02:11:32*

**Authors:** Toshiro Takami

**Comments:** 15 Pages.

**Category:** Number Theory

[1208] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-03 04:41:09*

**Authors:** Toshiro Takami

**Comments:** 17 Pages.

**Category:** Number Theory

[1207] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-01 23:31:19*

**Authors:** Toshiro Takami

**Comments:** 13 Pages.

**Category:** Number Theory

[1206] **viXra:1910.0081 [pdf]**
*replaced on 2019-11-01 05:18:49*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

**Category:** Number Theory

[1205] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-30 23:02:10*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

**Category:** Number Theory

[1204] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-29 19:21:56*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

**Category:** Number Theory

[1203] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-25 20:38:14*

**Authors:** Toshiro Takami

**Comments:** 15 Pages.

**Category:** Number Theory

[1202] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-25 03:15:35*

**Authors:** Toshiro Takami

**Comments:** 16 Pages.

**Category:** Number Theory

[1201] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-24 03:13:36*

**Authors:** Toshiro Takami

**Comments:** 14 Pages.

**Category:** Number Theory

[1200] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-21 17:16:57*

**Authors:** Toshiro Takami

**Comments:** 14 Pages.

**Category:** Number Theory

[1199] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-21 00:26:58*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

**Category:** Number Theory

[1198] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-20 00:32:59*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

**Category:** Number Theory

[1197] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-19 22:46:51*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

**Category:** Number Theory

[1196] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-18 18:42:32*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

**Category:** Number Theory

[1195] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-18 01:23:40*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

**Category:** Number Theory

[1194] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-16 07:09:47*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1193] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-15 23:48:53*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1192] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-13 22:06:58*

**Authors:** Toshiro Takami

**Comments:** 19 Pages.

**Category:** Number Theory

[1191] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-12 03:17:17*

**Authors:** Toshiro Takami

**Comments:** 33 Pages.

**Category:** Number Theory

[1190] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-10 23:42:33*

**Authors:** Toshiro Takami

**Comments:** 33 Pages.

**Category:** Number Theory

[1189] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-10 05:09:13*

**Authors:** Toshiro Takami

**Comments:** 16 Pages.

**Category:** Number Theory

[1188] **viXra:1910.0081 [pdf]**
*replaced on 2019-10-07 01:35:36*

**Authors:** Toshiro Takami

**Comments:** 17 Pages.

**Category:** Number Theory

[1187] **viXra:1910.0017 [pdf]**
*replaced on 2019-10-02 05:02:19*

[1186] **viXra:1909.0655 [pdf]**
*replaced on 2019-10-22 16:43:33*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. Proof for Riemann hypothesis and explanations for manifested properties of two types of Gram points.

"Mathematics for Incompletely Predictable Problems" makes all mathematical arguments valid and complete in this current paper (based on first key step of converting Riemann zeta function into its continuous format version) and our next paper (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 perfectly 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

[1185] **viXra:1909.0655 [pdf]**
*replaced on 2019-10-21 00:09:20*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. Proving Riemann hypothesis and explaining manifested properties of both Gram points

"Mathematics for Incompletely Predictable Problems" makes all mathematical arguments valid and complete in this current paper (based on first key step of converting Riemann zeta function into its continuous format version) and our next paper (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 perfectly 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

[1184] **viXra:1909.0654 [pdf]**
*replaced on 2019-10-23 06:25:42*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. Proofs for Polignac's and Twin prime conjectures.

"Mathematics for Incompletely Predictable Problems" makes all mathematical arguments valid and complete in our previous paper (based on first key step of converting Riemann zeta function into its continuous format version) and this current paper (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 perfectly 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

[1183] **viXra:1909.0654 [pdf]**
*replaced on 2019-10-22 16:37:36*

**Authors:** John Yuk Ching Ting

**Comments:** 21 Pages. Proofs for Polignac's and Twin prime conjectures.

"Mathematics for Incompletely Predictable Problems" makes all mathematical arguments valid and complete in our previous paper (based on first key step of converting Riemann zeta function into its continuous format version) and this current paper (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 perfectly 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

[1182] **viXra:1909.0654 [pdf]**
*replaced on 2019-10-21 00:13:22*

**Authors:** John Ting

**Comments:** 21 Pages. Proving Polignac's and Twin prime conjectures.

"Mathematics for Incompletely Predictable Problems" makes all mathematical arguments valid and complete in our previous paper (based on first key step of converting Riemann zeta function into its continuous format version) and this current paper (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 perfectly 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

[1181] **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

[1180] **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

[1179] **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

[1178] **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

[1177] **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

[1176] **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

[1175] **viXra:1909.0385 [pdf]**
*replaced on 2019-09-24 18:52:15*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1174] **viXra:1909.0385 [pdf]**
*replaced on 2019-09-22 00:51:56*

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

**Category:** Number Theory

[1173] **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

[1172] **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

[1171] **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

[1170] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-20 03:32:04*

**Authors:** Toshiro Takami

**Comments:** 37 Pages.

**Category:** Number Theory

[1169] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-19 03:16:57*

**Authors:** Toshiro Takami

**Comments:** 35 Pages.

**Category:** Number Theory

[1168] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-17 08:58:39*

**Authors:** Toshiro Takami

**Comments:** 38 Pages.

**Category:** Number Theory

[1167] **viXra:1909.0165 [pdf]**
*replaced on 2019-11-02 02:24:43*

**Authors:** Sitangsu Maitra

**Comments:** 6 pages

Proof of Goldbach's strong conjecture in a different way

**Category:** Number Theory

[1166] **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

[1165] **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

[1164] **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

[1163] **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

[1162] **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

[1161] **viXra:1909.0027 [pdf]**
*replaced on 2019-11-24 16:51:28*

**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