[11] **viXra:1810.0288 [pdf]**
*submitted on 2018-10-19 02:09:09*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

If prime, we found that the remainder is a prime number including 1 when divided by 30, 60 and 90.
This is called toshichan-man's small theorem.
It seems to be useful for prime number determination (especially huge prime number determination).

**Category:** Number Theory

[10] **viXra:1810.0282 [pdf]**
*submitted on 2018-10-17 06:31:37*

**Authors:** Vyacheslav Telnin

**Comments:** 116 Pages. format A5

In this paper, we analyze the construction of numbers from natural to complex. A close relationship between these numbers and three direct operations ([1] – addition, [2] – multiplication, [3] – exponentiation) is revealed. A method of constructing new direct operations is found. Two new direct operations ([4] and [0]) are constructed. Two inverse operations are constructed for each of them. If equations that are unsolvable in complex numbers are found for them, new numbers can be constructed on the basis of these unsolvabilities.
So far in this way in this work the question numbers are constructed (?- numbers) for [3] direct operation.
Along the way, numbers-strings and N-numbers are described. The topology of the numerical line is traced on the basis of N-numbers.
In this paper we find the relation of N-numbers and ?-numbers'.

**Category:** Number Theory

[9] **viXra:1810.0255 [pdf]**
*submitted on 2018-10-16 17:40:35*

**Authors:** Zeolla Gabriel Martín

**Comments:** 3 Pages.

This paper develops the construction of the Golden Pattern for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. These patterns form the sequence of prime numbers.

**Category:** Number Theory

[8] **viXra:1810.0240 [pdf]**
*submitted on 2018-10-15 14:59:35*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I found out my original make prime number built method.
30n+7 (n is positive integer, including 0).
30n+17 (n is positive integer, including 0).
Prime has a period of 30. We focused on that point and developed it.
However, those that are not prime are also quite included.
And,
30n+11 (n is positive integer, including 0).
30n+13 (n is positive integer, including 0).
Also considered.

**Category:** Number Theory

[7] **viXra:1810.0223 [pdf]**
*submitted on 2018-10-14 17:11:04*

**Authors:** Marco Ripà

**Comments:** 4 Pages.

In 2011, in his book “La strana coda della serie n^n^...^n", M. Ripà analyzed some properties involving the rightmost figures of integer tetration, the iterated exponentiation a^^b, characterized by an increasing number of stable digits for any base a > 1.
A few conjectures arose about how many new stable digits are generated by any unitary increment of the hyperexponent b, and Ripà indicated this value as V(a) or “convergence speed” of a. In fact, when b is large enough, V(a) seems to not depend from b, taking on a (strictly positive) unique value, and many observations supported this claim. Moreover, we claim that V(a) = 1 for any a(mod 25) congruent to {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a)>=2 otherwise.

**Category:** Number Theory

[6] **viXra:1810.0215 [pdf]**
*submitted on 2018-10-13 16:36:02*

**Authors:** Toshiro Takami

**Comments:** 2 Pages.

I calculated ζ (13) and ζ(15).
ζ (13) and ζ(15) tended to converge very quickly.

**Category:** Number Theory

[5] **viXra:1810.0175 [pdf]**
*submitted on 2018-10-12 02:45:55*

**Authors:** Anna Povazanova, Ivo Povazan

**Comments:** 13 Pages.

p is prime.The article describes the new Abelian groups of type
p=4k+1 and p = 4k-1, for which a theorem similar to the Fermat's
little theorem applies. The multiplicative group (Z/pZ)* in some sense
similar to the Abelian group of type p = 4k+1. Abelian group of type
p = 4k-1 is a different structure compared to group (Z/pZ)*. This
fact is used for the primality test of integer N = 4k-1. The primality
test was veried up to N = 2^(64).

**Category:** Number Theory

[4] **viXra:1810.0153 [pdf]**
*submitted on 2018-10-09 07:38:30*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents an integral for Catalan's constant: G=0.915965...

**Category:** Number Theory

[3] **viXra:1810.0141 [pdf]**
*submitted on 2018-10-09 16:57:44*

**Authors:** Jonathan Trousdale

**Comments:** 6 Pages.

This paper sets forth a representation of the hyperbolic substratum that defines order on $\mathbb{N}$. Degeneracy of $\mathbb{N}$ at points of intersection with the substratum is observed as violations of the fundamental theorem of arithmetic in the form of mutable prime factorization. At a point of maximum symmetry on the representation manifold, an exact expression of $\pi$ is available as a combination of three integers.

**Category:** Number Theory

[2] **viXra:1810.0108 [pdf]**
*replaced on 2018-10-07 19:41:16*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

I calculated ζ (17) and ζ(19).
ζ (17) and ζ(19) tended to converge very quickly.

**Category:** Number Theory

[1] **viXra:1810.0046 [pdf]**
*submitted on 2018-10-05 01:33:59*

**Authors:** Idriss Olivier Bado

**Comments:** 5 Pages.

The main contribution of this paper is to achieve the proof of Riemann hypothesis. The key idea is based on new formulation of the problem
$$\zeta(s)=\zeta(1-s) \Leftrightarrow re(s)=\frac{1}{2}$$. This proof is considered as a great discovery in mathematic.

**Category:** Number Theory