# Number Theory

## 1901 Submissions

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

### Definitive Proof of Legendre's Conjecture

Authors: Kenneth A. Watanabe

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

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

### A Note About the Abc Conjecture a Proof of the Conjecture: C 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

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

### Proof of Goldbach Conjecture 2019-01-06

Authors: Michael Grützmann

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

 viXra:1901.0227 [pdf] replaced on 2019-12-27 16:44:32

### Collatz Conjecture Proof

Authors: James Edwin Rock

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

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

### Elements 5 : Three Trigonometric Identities

Authors: Edgar Valdebenito

This note presents three trigonometric identities.
Category: Number Theory

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

### A Parametric Equation of the Equation A^5 + B^5 = 2c^2

Authors: Quang Nguyen Van

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

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

### Assuming ABC Conjecture is True Implies Beal Conjecture is True

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

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

### A Resolution Of The Brocard-Ramanujan Problem

Authors: Johnny E. Magee

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

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

### Relation Between the Euler Totient, the Counting Prime Formula and the Prime Generating Functions

Authors: Nazihkhelifa

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

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

### Comparison of the Theoretical and Empirical Results for the Benford's Law Summation Test Performed on Data that Conforms to a Log Normal Distribution

Authors: Robert C. Hall