Number Theory

1208 Submissions

[5] 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
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

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

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

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
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

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

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

Irrationality of the Euler-Mascheroni Constant

Authors: Andile Mabaso
Comments: 6 Pages.

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