[11] **viXra:1209.0100 [pdf]**
*submitted on 2012-09-27 20:49:05*

**Authors:** Tian-Chou Wang

**Comments:** 40 Pages. Manuscript scan, in Chinese, and copyright belongs to the author

This is a part of Dr. Tian-Chou Wang's proof to the Riemann Hypothesis. It follows 'the opening' (viXra citation number: 1209.0063), Chapter 1 (viXra citation number: 1209.0068), Chapter 2 (viXra citation number: 1209.0078), Chapter 3 (viXra citation number: 1209.0090).

**Category:** Number Theory

[10] **viXra:1209.0099 [pdf]**
*replaced on 2013-06-02 11:53:37*

**Authors:** Michael Pogorsky

**Comments:** 7 Pages.

The Fermat’s Last Theorem is proved by means of general algebra in four major steps. a)The expressions for a, b, c of type a=uwv+v^n;b=uwv+w^n; c=uwv+v^n+w^n required to satisfy equation a^n+b^n=c^n deduced for two main versions of the equation. b)The existence of positive integers u_p and c_p such that a+b is divided by u_p^n and c is divided by c_p u_p proved to be required. c)Polynomial a^n+b^n presented through expressions for a and b proved to be a sum of three divisible by c polynomials. d)The long division of one of them w^(n∙n)+v^(n∙n) by either of two other gives remainder not divisible by c. This contradiction proves the Theorem.

**Category:** Number Theory

[9] **viXra:1209.0094 [pdf]**
*replaced on 2013-02-24 21:31:35*

**Authors:** Talon J. Ward

**Comments:** 8 Pages.

The Collatz conjecture is a famous problem in number theory. Given an integer, if it's odd, multiply it by three and add one, or, if it's even, divide it by two. The Collatz conjecture states that any trajectory of iterates of this Collatz transformation on the positive integers will reach one in a finite number of steps. This problem explores the behavior of a complicated discrete dynamical system that has eluded solution for over seventy years.

This paper addresses the Collatz conjecture by altering the Collatz transformation into a friendlier format, which tells us what to do with an odd integer given its congruence modulo eight. We then describe how to find the numbers whose first few iterates follow a given pattern, which leads us to a directed graph that every trajectory must eventually enter. This directed graph then shows us that, in a finite number of steps, every iterate of a trajectory must either converge to one or strictly increase thereafter. Since there is no number whose trajectory strictly increases, the Collatz conjecture holds.

**Category:** Number Theory

[8] **viXra:1209.0090 [pdf]**
*submitted on 2012-09-25 19:08:59*

**Authors:** Tian-Chou Wang

**Comments:** 41 Pages. Manuscript scan, in Chinese, and copyright belongs to the author.

This is a part of Dr. Tian-Chou Wang's proof to the Riemann Hypothesis. It follows 'the opening' (viXra citation number: 1209.0063), Chapter 1 (viXra citation number: 1209.0068), and Chapter 2 (viXra citation number: 1209.0078).

**Category:** Number Theory

[7] **viXra:1209.0079 [pdf]**
*replaced on 2012-09-27 10:50:01*

**Authors:** Ibrahim M. Alabdulmohsin

**Comments:** 135 Pages.

In this manuscript, we present the foundations of Summability
Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite differences under a single simple umbrella. Using Summability Calculus, any given finite sum bounded by a variable n becomes immediately in analytic form. Not only can we differentiate and integrate with respect to the bound n without having to rely on an explicit analytic formula for the finite sum, but we can also deduce asymptotic expansions, accelerate convergence, assign natural values to divergent sums, and evaluate the finite sum for any complex value of n. This follows because the discrete definition of the simple finite sum embodies a unique natural continuation to the entire complex plane. Throughout the paper, many established results are strengthened such as the Bohr-Mollerup theorem, Stirling's approximation, Glaisher's approximation, and the Shannon-Nyquist sampling theorem. In addition, many celebrated theorems are extended and generalized such as the Euler- Maclaurin summation formula and Boole's summation formula. Finally, we show that countless identities that have been proved throughout the past 300 years by different mathematicians using different approaches can actually be derived in an elementary straightforward manner using the rules of Summability Calculus.

**Category:** Number Theory

[6] **viXra:1209.0078 [pdf]**
*submitted on 2012-09-23 19:30:41*

**Authors:** Tian-Chou Wang

**Comments:** 33 Pages. Manuscript scan, in Chinese, and copyright belongs to the author.

Chapter II is a part of a series of Dr. Tian-Chou Wang's proof to the Riemann Hypothesis.
It follows 'the opening' (viXra citation number: 1209.0063) and Chapter 1 (viXra citation number: 1209.0068).

**Category:** Number Theory

[5] **viXra:1209.0068 [pdf]**
*submitted on 2012-09-20 23:54:14*

**Authors:** Tian-Chou Wang

**Comments:** 13 Pages. Publication scan, in Chinese, and copyright belongs to the author.

Chapter I is a part of a series of publications by Dr. Tian-Chou Wang, who has tested the Riemann Hypothesis. It follows 'the opening' (viXra citation number: 1209.0063).

**Category:** Number Theory

[4] **viXra:1209.0063 [pdf]**
*submitted on 2012-09-19 20:02:11*

**Authors:** Tian-Chou Wang

**Comments:** 41 Pages. Manuscript scan, in Chinese, and copyright belongs to the author.

This is a series of publication by Dr. Tian-Chou Wang, who tested the Riemann Hypothesis. His proof contains eleven chapters and eleven appendices, and is composed of about 500,000words and 5000 equations in total.
The first manuscript of 'the opening' provides the overview of Riemann Hypothesis and the work flow of Dr. Wang's proof.

**Category:** Number Theory

[3] **viXra:1209.0036 [pdf]**
*submitted on 2012-09-12 19:27:41*

**Authors:** Chun-Xuan Jiang

**Comments:** 8 Pages.

中国人把费马大定理成果送绐Wiles是犯罪

**Category:** Number Theory

[2] **viXra:1209.0032 [pdf]**
*submitted on 2012-09-11 14:28:10*

**Authors:** Stephen Crowley

**Comments:** 4 Pages.

The Frobenius-Perron transfer operator of the harmonic sawtooth map is investigated and
some expressions for its eigenvalues are found.

**Category:** Number Theory

[1] **viXra:1209.0006 [pdf]**
*submitted on 2012-09-03 06:41:39*

**Authors:** Vladimir Godovalov

**Comments:** 12 Pages. Russian version

This paper is supplement to Powers Fields Theory, http://vixra.org/abs/1206.0025. Here in more comprehensive way the substance of Fermat’s last theorem is revealed. It becomes possible, because in fact, it is closely related to Pascal’s triangle so its proof is achieving for all the exponents without extreme difficulties.

**Category:** Number Theory