[34] **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

[33] **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

[32] **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

[31] **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

[30] **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

[29] **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

[28] **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

[27] **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

[26] **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

[25] **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

[24] **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

[23] **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

[22] **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

[21] **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

[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **viXra:1909.0334 [pdf]**
*replaced on 2019-12-11 23:42:01*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 3 Pages.

In this paper, we propose the axiomatic regularity of prime numbers.

**Category:** Number Theory

[14] **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

[13] **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

[12] **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

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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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