[12] **viXra:1901.0436 [pdf]**
*replaced on 2019-06-13 16:45:16*

**Authors:** Kenneth A. Watanabe

**Comments:** 19 Pages.

Legendre's conjecture, states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that accurately determines the number of prime numbers less than n for large values of n. Then, using this equation, it was proven by induction that there is at least one prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture for sufficiently large values n. The error between the derived equation and the actual number of prime numbers less than n was empirically proven to be very small (0.291% at n = 50,000), and it was proven that the size of the error declines as n increases, thus validating the proof.

**Category:** Number Theory

[11] **viXra:1901.0430 [pdf]**
*replaced on 2019-01-31 12:44:22*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. Paper corrected of a mistake in the last version. Comments welcome.

In this paper, we consider the abc conjecture, then we give a proof of the conjecture c<rad^2(abc) that it will be the key to the proof of the abc conjecture.

**Category:** Number Theory

[10] **viXra:1901.0300 [pdf]**
*submitted on 2019-01-19 14:41:09*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

On calculation by Euler 's formula(3), at least notice that the zero point of the real part is not on x = 0.5.
Using Euler 's formula, we found that at least the real part' s zero point is not x = 0.5 but about x = 0.115444. Moreover, the imaginary point is around i14.524.
And s=0.8355 +i39.
And s=0.1645 +i39.
And s=0.884556 +i14.524.
And s=0.115444 +i14.524
Also, replacing sin with cos, the imaginary part becomes zero.
I do not know at all whether the collapse of Riemann hypothesis or not?
In addition, books are printed as cos instead of sin.
Also, I have collected ζ on the left side.
In addition, (8) is Euler's formula found overseas, which is also a singular point in this. Also, I have collected ζ on the left side.
In this case, sin is printed instead of cos, but if sin and cos are exchanged, the zero point moves only from the real part to the imaginary part.

**Category:** Number Theory

[9] **viXra:1901.0297 [pdf]**
*submitted on 2019-01-19 22:35:49*

**Authors:** Michael Grützmann

**Comments:** 2 Pages.

every prime number can be a sum of p=3+...+3+2 or
q=3+...+3+4, the number of '3's in both equatons always being odd.
there are the for two primes p+q the combinations
'p+q', 'p+p' and 'q+q'.
we consider case 1: p+q=3k+2+3l+4
3k must be odd, as the product of two odd numbers,
3l must be odd, for the same reason.
but the sum of two odd numbers is an even number always. also, if you add more even numbers, like 2 and 4, the result will always be even also.
So this results in an even number.
cases 'p+p' and 'q+q' analogue.

**Category:** Number Theory

[8] **viXra:1901.0227 [pdf]**
*replaced on 2019-06-20 11:56:51*

**Authors:** James Edwin Rock

**Comments:** 12 Pages.

Collatz sequences are formed by applying the Collatz algorithm to any positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. If the Collatz conjecture is true eventually you always get back to one. A connected Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[7] **viXra:1901.0155 [pdf]**
*submitted on 2019-01-11 06:22:54*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents three trigonometric identities.

**Category:** Number Theory

[6] **viXra:1901.0116 [pdf]**
*submitted on 2019-01-10 02:36:15*

**Authors:** Quang Nguyen Van

**Comments:** 2 Pages.

The equation a^5 + b^5 = c^2 has no solution in integer. However, related to Fermat- Catalan conjecture, the equation a^5 + b^5 = 2c^2 has a solution in integer. In this article, we give a parametric equation of the equation a^5 + b^5 = 2c^2.

**Category:** Number Theory

[5] **viXra:1901.0108 [pdf]**
*submitted on 2019-01-08 11:13:12*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. A Proof of ABC conjecture is submitted to the Journal of Number Theory (2019). Comments Welcome.

In this paper, we assume that the ABC conjecture is true, then we give a proof that Beal conjecture is true. We consider that Beal conjecture is false then we arrive to a contradiction. We deduce that the Beal conjecture is true.

**Category:** Number Theory

[4] **viXra:1901.0101 [pdf]**
*submitted on 2019-01-09 00:16:39*

**Authors:** Johnny E. Magee

**Comments:** 7 Pages.

We identify equivalent restatements of the Brocard-Ramanujan diophantine equation, $(n! + 1) = m^2$; and employing the properties and implications of these equivalencies, prove that for all $n > 7$, there are no values of $n$ for which $(n! + 1)$ can be a perfect square.

**Category:** Number Theory

[3] **viXra:1901.0046 [pdf]**
*submitted on 2019-01-04 11:35:20*

**Authors:** Nazihkhelifa

**Comments:** 4 Pages.

Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
The theory of numbers is an area of mathematis hiih eals ith
the propertes of hole an ratonal numbers... In this paper I ill
intro uie relaton bet een Euler phi funiton an prime iountnn
an neneratnn formula, as ell as a ioniept of the possible
operatons e ian use ith them. There are four propositons hiih
are mentone in this paper an I have use the efnitons of these
arithmetial funitons an some Lemmas hiih refeit their
propertes, in or er to prove them

**Category:** Number Theory

[2] **viXra:1901.0030 [pdf]**
*submitted on 2019-01-03 17:17:16*

**Authors:** Robert C. Hall

**Comments:** 24 Pages.

The Benford's Law Summation test consists of adding all numbers that begin with a particular first or first two digits and determining its distribution with respect to these first or first two digits numbers. Most people familiar with this test believe that the distribution is a uniform distribution for any distribution that conforms to Benford's law i.e. the distribution of the mantissas of the logarithm of the data set is uniform U[0,1). The summation test that results in a uniform distribution is true for an exponential function (geometric progression) but not true for a data set that conforms to a Log Normal distribution even when the Log Normal distribution itself closely approximates a Benford's Law distribution.

**Category:** Number Theory

[1] **viXra:1901.0007 [pdf]**
*replaced on 2019-06-04 05:20:45*

**Authors:** Rachid Marsli

**Comments:** 17 Pages.

In this work, the author shows a sufficient and necessary condition for an integer of the form (z^n-y^n)/(z-y) to be divisible by some perfect mth power p^m,where p is an odd prime and m is a positive integer. A constructive method of this type of integers is explained with details and examples. Links beetween the main result and known ideas such as Fermat’s last theorem, Goor-maghtigh conjecture and Mersenne numbers are discussed. Other relatedideas, examples and applications are provided.

**Category:** Number Theory