# Number Theory

## 1208 Submissions

 viXra:1208.0245 [pdf] replaced on 2013-11-16 10:16:33

### The Arithmetic of Binary Representations of Even Positive Integer 2n and Its Application to the Solution of the Goldbach's Binary Problem

Authors: Alexander Fedorov

One of causes why Goldbach's binary problem was unsolved over a long period is that binary representations of even integer 2n (BR2n) in the view of a sum of two odd primes(VSTOP) are considered separately from other BR2n. By purpose of this work is research of connections between different types of BR2n. For realization of this purpose by author was developed the "Arithmetic of binary representations of even positive integer 2n" (ABR2n). In ABR2n are defined four types BR2n. As shown in ABR2n all types BR2n are connected with each other by relations which represent distribution of prime and composite positive integers less than 2n between them. On the basis of this relations (axioms ABR2n) are deduced formulas for computation of the number of BR2n (NBR2n) for each types. In ABR2n also is defined and computed Average value of the number of binary sums are formed from odd prime and composite positive integers $< 2n$ (AVNBS). Separately AVNBS for prime and AVNBS for composite positive integers. We also deduced formulas for computation of deviation NBR2n from AVNBS. It was shown that if $n$ go to infinity then NBR2n go to AVNBS that permit to apply formulas for AVNBS to computation of NBR2n. At the end is produced the proof of the Goldbach's binary problem with help of ABR2n. For it apply method of a proof by contradiction in which we make an assumption that for any 2n not exist BR2n in the VSTOP then make computations at this conditions then we come to contradiction. Hence our assumption is false and forall $2n > 2$ exist BR2n in the VSTOP.
Category: Number Theory

 viXra:1208.0022 [pdf] replaced on 2014-04-03 22:09:17

### On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures

Authors: Germán Paz
Comments: 11 Pages. The title and the abstract have been modified; a few references have been added, as it has been suggested to the author. This paper is also available at arxiv.org/abs/1310.1323.

Let $n\in\mathbb{Z}^+$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number? In this paper we show that this is actually the case for every $n \leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.
Category: Number Theory

 viXra:1208.0021 [pdf] submitted on 2012-08-06 08:44:18

### Smarandache 函数 及其相关问题研究 Vol.8 / Research on Smarandache Functions and Other Related Problems, Vol. 8 [in Chinese Language Only]

Authors: Wang Tingting, Liu Yanni

Category: Number Theory

 viXra:1208.0011 [pdf] submitted on 2012-08-03 19:23:15

### Two New Constants \niu and \theta and a New Formula \pi = (1/2)e^\theta

Authors: Chen Wenwei