[18] **viXra:1710.0209 [pdf]**
*submitted on 2017-10-18 15:14:30*

**Authors:** Leif R. Uppström, Daniel Uppström

**Comments:** 9 Pages.

In mathematical literature it is asked for a computable function or efficient algorithm to find all, or at least a large subset, of the prime numbers. This paper shows that all primes can be characerised by their reciprocal period length *L* and its figure value *R*. These parameters are given for each prime after inversion to an infinitely repeated period and are used to group all primes into disjoint sets that arise as a function of a geometric progression. This theory suggests new ways to enumerate and find large primes.

**Category:** Number Theory

[17] **viXra:1710.0205 [pdf]**
*submitted on 2017-10-19 02:50:41*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

In this paper it is proved that the sum of consecutive prime numbers under the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, it is proved another asymptotic relationship between the sum of the first prime numbers up to the integer part of the square root of a given natural number and the prime counting function.

**Category:** Number Theory

[16] **viXra:1710.0202 [pdf]**
*submitted on 2017-10-15 09:49:16*

**Authors:** I. N. Tukaev

**Comments:** 2 Pages.

A recurrence relation is defined for calculating a sequence of prime numbers using the Riemann zeta function and the Euler product.

**Category:** Number Theory

[15] **viXra:1710.0174 [pdf]**
*submitted on 2017-10-17 01:35:25*

**Authors:** Choe Ryujin

**Comments:** 4 Pages.

Theorem of prime pair distribution

**Category:** Number Theory

[14] **viXra:1710.0170 [pdf]**
*submitted on 2017-10-17 09:12:45*

**Authors:** José Francisco García Juliá

**Comments:** 2 Pages.

It is obtained a solution of the Fermat conjecture.

**Category:** Number Theory

[13] **viXra:1710.0169 [pdf]**
*submitted on 2017-10-17 09:44:37*

**Authors:** Steven Shawcross

**Comments:** 9 Pages. A version of this paper is copyrighted by Steven Shawcross, 2003.

The integer 2 satisfies the divisibility definition of a prime number: it is only divisible by itself and 1. The integer 1 also satisfies this definition, and yet, mathematicians generally do not consider 1 a prime. Rather 1 merits a class of its own, belonging neither to the prime nor composite class. In divisibility theory, 2 does occupy a special subclass within the class of prime numbers: it is the only even prime. This paper introduces a theory of numbers called the Prime Set Representation Theory. This theory utilizes the odd primes and does not rely on the primeness of 2. In Prime Set Representation Theory, the odd primes are building blocks of the theory; all integers, including 2, have representations in terms of them. The import of the theory is not to dislodge the integer 2 from its solitary, even-prime status. The theory's efficacy is a better understanding of the distribution of primes, twin primes, and primes of the form x^2 + 1. A natural extension of the theory yields valid and strikingly direct approximation formulas for these prime classifications. The same theory furnishes a new and improved approximation to the number of Goldbach pairs associated with general even number 2n (the improvement is relative to Sylvester's formula for Goldbach pairs, but the formula performs well vis-à-vis the Hardy-Littlewood formulas in the ranges tested).

**Category:** Number Theory

[12] **viXra:1710.0145 [pdf]**
*submitted on 2017-10-12 05:04:05*

**Authors:** Timothy W. Jones

**Comments:** 19 Pages.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(2n+1), n>1, is irrational.

**Category:** Number Theory

[11] **viXra:1710.0129 [pdf]**
*submitted on 2017-10-11 11:54:50*

**Authors:** Robert C. Hall

**Comments:** 28 Pages.

Regarding Benford's law, many believe that the statistical data sources follow a Benford's law probability density function(1/xLn(10))when, in actuality, it follows a Lognormal probability density function. The only data that strictly follows a Benford's law probability density function is an exponential function i.e. a number (base) raised to a power x. The other sets of data conform to a Lognormal distribution and, as the standard deviation approaches infinity, approximates a true Benford distribution.
Also, the so called Summation theorem whereby the sum of the values with respect to the first digits is a uniform distribution only applies to an exponential function. The data derived from the aforementioned Lognormal distribution is more likely to conform to a Benford like distribution as the data seems to indicate.

**Category:** Number Theory

[10] **viXra:1710.0113 [pdf]**
*submitted on 2017-10-10 06:32:08*

**Authors:** Kurmet Sultan

**Comments:** 2 Pages. This is the Russian version of the manuscript.

The paper describes the First theorem of Kurmet and a simple proof of the Last theorem of Fermat, which was obtained on the basis of Kurmet's First Theorem.

**Category:** Number Theory

[9] **viXra:1710.0112 [pdf]**
*submitted on 2017-10-10 06:35:55*

**Authors:** Kurmet Sultan

**Comments:** 2 Pages. This is the Russian version of the manuscript.

In this paper we describe the Second Theorem of Kurmet and give a simple proof of Catalan’s conjecture on the basis of Kurmet's Second Theorem.

**Category:** Number Theory

[8] **viXra:1710.0109 [pdf]**
*submitted on 2017-10-09 03:05:33*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Fermats Last Theorem n=4
One line proof

**Category:** Number Theory

[7] **viXra:1710.0099 [pdf]**
*submitted on 2017-10-10 01:50:30*

**Authors:** Zhang Tianshu

**Comments:** 19 Pages.

We first classify all integers ≥2 into eight kinds, and that formulate each of seven kinds therein into a sum of three unit fractions. For remainder one kind, we classify it into three genera, and that formulate each of two genera therein into a sum of three unit fractions. For remainder one genus, we classify it into five sorts, and that formulate each of three sorts therein into a sum of three unit fractions. For remainder two sorts i.e. 4/(49+120c) and 4/(121+120c) with c≥0, we prove them by logical inference. But miss out 3587 concrete fractions to await computer programming to solve the problem that express each of them into a sum of three unit fractions.

**Category:** Number Theory

[6] **viXra:1710.0072 [pdf]**
*submitted on 2017-10-07 15:48:59*

**Authors:** Henry L. Mitchell

**Comments:** Pages. please remove comment

We introduce a sieve for the number of twin primes less than N by sieving through the set {k ∊ ℤ+ | 6k < N}. We derive formula accordingly using the Euler product and the Brun. Sieve.
We then use the Prime Number Theorem and Mertens’ Theorem.
The main results are:
1) A sieve for the twin primes similar to the sieve of Eratosthenes for primes involving only the
values of k, the indices of the multiples of 6, ranging over k = p ,5 ≤ p <√N
2) A formula for the approximate number of twin primes less than N in terms of the number of
primes less than N
3) The asymptotic formula for the number of twin primes less than N verifying the Hardy
Littlewood Conjecture.

**Category:** Number Theory

[5] **viXra:1710.0048 [pdf]**
*submitted on 2017-10-04 20:55:25*

**Authors:** Choe Ryujin

**Comments:** 1 Page.

Proof of Riemann hypothesis

**Category:** Number Theory

[4] **viXra:1710.0042 [pdf]**
*submitted on 2017-10-03 11:01:23*

**Authors:** Dieter Sengschmitt

**Comments:** 15 Pages.

I can proof that there are infinitely many twin primes. The twin prime counting function π2(n), which gives the number of twin primes less than or equal to n for any natural number n, is for
limn→∞ π2(n)= 2 C2 [π(n)]^2/n
where π(n) is the prime counting function and C2 is the so-called twin prime constant with C2=0,6601618…

**Category:** Number Theory

[3] **viXra:1710.0038 [pdf]**
*submitted on 2017-10-03 16:37:46*

**Authors:** Robert C. Hall

**Comments:** 2 Pages.

An attempt is made to derive the probability density function of the sum of prime numbers, which is x/Ln(x). This does appear to be quite accurate in predicting the sum of prime numbers less than 100,000( within 0.124%). Given this assertion, an attempt is made to derive the probability density function of the distribution of the prime numbers themselves.

**Category:** Number Theory

[2] **viXra:1710.0017 [pdf]**
*submitted on 2017-10-02 02:24:57*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

four new formulas for pi

**Category:** Number Theory

[1] **viXra:1710.0015 [pdf]**
*submitted on 2017-10-02 03:09:57*

**Authors:** Lulu Karami

**Comments:** 4 Pages.

This submission gives a closed form identity similar to one given by
Ramanujan. A formula for infinitely many similar identities is presented
here as well.

**Category:** Number Theory