[19] **viXra:1705.0360 [pdf]**
*replaced on 2017-05-25 07:46:44*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

This Identity proofs direct Fermats Last Theorem

**Category:** Number Theory

[18] **viXra:1705.0343 [pdf]**
*submitted on 2017-05-22 13:32:15*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas for pi constant

**Category:** Number Theory

[17] **viXra:1705.0342 [pdf]**
*submitted on 2017-05-22 13:36:50*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

This note presents some formulas related with the number z=LambertW(i),where LambertW(x) is the Lambert function.

**Category:** Number Theory

[16] **viXra:1705.0293 [pdf]**
*submitted on 2017-05-19 11:18:33*

**Authors:** Prashanth Rao

**Comments:** 1 Page.

If p is any odd prime number and c is any odd number less than p, then there must exist a positive number c’ less than p, such that cc’= -2modp

**Category:** Number Theory

[15] **viXra:1705.0289 [pdf]**
*submitted on 2017-05-19 07:56:37*

**Authors:** Helmut Preininger

**Comments:** 14 Pages.

In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions.
We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.

**Category:** Number Theory

[14] **viXra:1705.0277 [pdf]**
*submitted on 2017-05-19 01:01:55*

**Authors:** Shaban A. Omondi Aura

**Comments:** 30 Pages. Preferably for journals, academies and conferences

This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approximates the number of prime numbers at a given magnitude, from N=3 to N=〖10〗^25. Concerning the location of an nth prime number, the paper has devised a method that can help us locate a given prime number within specified bounds. Nonetheless, the paper has formulated an equation that can help us determine extremely bounded gaps. Lastly, using trans-algebraic number theory method, the paper has shown that unpredictable behaviors of prime numbers are due to their identity nature.

**Category:** Number Theory

[13] **viXra:1705.0242 [pdf]**
*submitted on 2017-05-16 03:47:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any pair of twin primes [p, q], p ≥ 11, there exist a number n having the sum of its digits equal to 12 such that inserting n after the first digit of p respectively q are obtained two primes (almost always twins, as in the case [1481, 1483] where n = 48 is inserted in [11, 13], beside the case that the first digit of twins is different, as in the case [5669, 6661] where n = 66 is inserted in [59, 61]).

**Category:** Number Theory

[12] **viXra:1705.0224 [pdf]**
*submitted on 2017-05-15 02:29:14*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 7, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 before the last digit of p.

**Category:** Number Theory

[11] **viXra:1705.0221 [pdf]**
*submitted on 2017-05-15 03:41:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p ≥ 5, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 after the first digit of p.

**Category:** Number Theory

[10] **viXra:1705.0154 [pdf]**
*submitted on 2017-05-09 12:16:42*

**Authors:** Mesut Kavak

**Comments:** 1 Page.

This works aims to bring a simple solution to the Riemann Hypothesis over the Lagarias Transformation.

**Category:** Number Theory

[9] **viXra:1705.0152 [pdf]**
*submitted on 2017-05-09 12:39:34*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas related with ahmed's integral.

**Category:** Number Theory

[8] **viXra:1705.0151 [pdf]**
*submitted on 2017-05-09 12:43:55*

**Authors:** Edgar Valdebenito

**Comments:** 19 Pages.

This note presents formulas related with Euler-Mascheroni constant and fractals.

**Category:** Number Theory

[7] **viXra:1705.0142 [pdf]**
*replaced on 2017-05-13 03:50:38*

**Authors:** Carlos Castro

**Comments:** 13 Pages. Submitted to Mod. Phys. Letts A

An approach to solving the Riemann Hypothesis is revisited within the framework of the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance. The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation
$ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $. A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$,
and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively.
A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.

**Category:** Number Theory

[6] **viXra:1705.0130 [pdf]**
*submitted on 2017-05-07 08:49:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of Poulet numbers P such that concatenating P to the left with the number (s(P) – 1)/2, where s is the sum of digits of P, is obtained a prime; also I make the same conjecture for (s(P) – 1)/3 respectively for (s(P) – 1)/6.

**Category:** Number Theory

[5] **viXra:1705.0129 [pdf]**
*submitted on 2017-05-07 08:51:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper, “Primes obtained concatenating a Poulet number P with (s - 1)/n where s digits sum of P and n is 2, 3 or 6”, I noticed that in almost all the cases that I considered if a prime was obtained through this concatenation than the digits sum of P was a prime. That gave me the idea for this paper where I observe that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1 (including 1 and s – 1).

**Category:** Number Theory

[4] **viXra:1705.0126 [pdf]**
*submitted on 2017-05-07 11:22:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper, “Primes obtained concatenating to the left a prime having an odd prime digit sum s with a divisor of s - 1”, I observed that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1. In this paper I conjecture that for any prime p, p ≠ 5, having an odd prime digit sum s there exist an infinity of primes obtained concatenating to the left p with multiples of s – 1. Yet I conjecture that there exist at least a prime obtained concatenating n*(s – 1) with p such that n < sqr s.

**Category:** Number Theory

[3] **viXra:1705.0117 [pdf]**
*replaced on 2017-05-22 09:41:44*

**Authors:** Jason Cole

**Comments:** 4 Pages.

There is exciting research trying to connect the nontrivial zeros of the Riemann Zeta function to Quantum mechanics as a breakthrough towards proving the 160-year-old Riemann Hypothesis. This research offers a radically new approach.
Most research up to this point have focused only on mapping the nontrivial zeros directly to eigenvalues. Those attempts have failed or didn’t yield any new breakthrough. This research takes a radically different approach by focusing on the quantum mechanical properties of the wave graph of Zeta as ζ(0.5+it) and not the nontrivial zeros directly. The conjecture is that the wave forms in the graph of the Riemann Zeta function ζ(0.5+it) is a wave function ψ. It is made of a Complex version of the Parity Operator wave function. The Riemann Zeta function consists of linked Even and Odd Parity Operator wave functions on the critical line. From this new approach, it shows the Complex version of the Parity Operator wave function is Hermitian and it eigenvalues matches the zeros of the Zeta function.

**Category:** Number Theory

[2] **viXra:1705.0115 [pdf]**
*submitted on 2017-05-05 16:53:48*

**Authors:** Christopher Goddard

**Comments:** 23 Pages.

Using an extension of the idea of the radical of a number, as well as a few other ideas, it is indicated as to why one might expect the Oesterle-Masser conjecture to be true. Based on structural elements arising from this proof, a criterion is then developed and shown to be potentially sufficient to resolve two relatively deep conjectures about the structure of the prime numbers. A sketch is consequently provided as to how it might be possible to demonstrate this criterion, borrowing ideas from information theory and cybernetics.

**Category:** Number Theory

[1] **viXra:1705.0100 [pdf]**
*submitted on 2017-05-03 13:05:47*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some double integrals for Euler-Mascheroni constant and related fractals.

**Category:** Number Theory