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

**Authors:** Alexander Fedorov

**Comments:** 50 Pages.

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

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

**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

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

**Authors:** Wang Tingting, Liu Yanni

**Comments:** 141 Pages.

本书主要介绍Smarandache 函数与伪Smarandache 函数、几类Smarandache 序列及其它相关问题的 最新研究进展, 同时还介绍了一些其它数论问题的研 究成果. 本书各部分内容编排相对独立,有兴趣的读者 可以从阅读本书任何一个章节开始, 开拓读者视野，激发读者对Smarandache 相关问题的研究兴趣.
This book mainly introduces the latest research progress of Smarandache functions and pseudo-Smarandache functions, several kinds of Smarandache sequences and other related problems. At the same time it also introduces some other study in number theory. This book commits itself to relatively independent content arrangement with each chapter and section, every reader who is interested in this book can read any chapter for the beginning. It could open up readers’ perspective; arouse readers to study Smarandache problems.

**Category:** Number Theory

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

**Authors:** Chen Wenwei

**Comments:** 6 Pages.

This paper brings to light two new constants and a formula.

**Category:** Number Theory

[1] **viXra:1208.0009 [pdf]**
*replaced on 2012-08-11 17:00:38*

**Authors:** Andile Mabaso

**Comments:** 6 Pages.

In this paper we prove that the Euler-Mascheroni constant is irrational and transcendental.

**Category:** Number Theory