# Number Theory

## 1209 Submissions

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

### The Answer to The Riemann Hypothesis - Chapter IV

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

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

### Proof of the Fermat's Last Theorem

Authors: Michael Pogorsky

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

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

### The Collatz Theorem

Authors: Talon J. Ward

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

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

### The Answer to The Riemann Hypothesis - Chapter III

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

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

### Summability Calculus

Authors: Ibrahim M. Alabdulmohsin

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

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

### The Answer to The Riemann Hypothesis - Chapter II

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

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

### The Answer to The Riemann Hypothesis - Chapter I

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

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

### The Answer to The Riemann Hypothesis - The Opening

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

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

### Chinese Put Fermat Last Theorem for Wiles is a Crime

Authors: Chun-Xuan Jiang

Category: Number Theory

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

### The Transfer Operator of the Harmonic Sawtooth Map

Authors: Stephen Crowley