Number Theory

The Perfect Cuboid Is Nothing More Than a Myth

Authors: Zakhar Pekhterev
Comments: 2 Pages. (Correction made by viXra Admin - Please conform!)

An impeccable proof of the impossibility of the existence of a perfect cuboid based on the parametrization of Leonhard Euler.
Category: Number Theory

Possible Solutions of the Rational Cuboid Problem

Authors: B. S. Safin
Comments: 5 Pages.

This article covers a number of forms for elliptic equations that were derived from the simultaneous equations describing a rational cuboid. The analysis of these elliptic equations shows that some rational points on the elliptic curves exist, but they are not the points of infinite order, accordingly they do not belong to any of the right triangles.
Category: Number Theory

Search for Congruent Numbers

Authors: B. S. Safin
Comments: 3 Pages.

By parametrizing the Pythagorean equation with hyperbolic functions you can obtain an algebraic equation of the 3rd degree that describes congruent numbers. In some cases this equation may facilitate the search for these numbers.
Category: Number Theory

The Elemental Property of Primes and Small Gaps Between Primes

Authors: Yung Zhao
Comments: 2 Pages.

The solution to the Twin Prime Conjecture lies in the elemental property of primes. We construct a sequence of consecutive primes, analyzing and handling them by the combination of the elemental property of primes and the Statistics theory reveal that Twin Prime Conjecture is true.
Category: Number Theory

On Non-Trivial Zeros and Riemann Zeta Function

Authors: Bertrand Wong
Comments: 5 Pages.

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]
Category: Number Theory

On Energy Numbers

Authors: Ryan J. Buchanan
Comments: 7 Pages.

The "energy numbers" of Parker Emmerson are critically examined, and an application of these exotic numbers to particle physics is attempted. Along the way, we establish the quaternion field identity, which is an isomorphism between a certain characterization of the abstract structure of a Hermitian space, and the complex Borel algebra of its generators.
Category: Number Theory

On the Set of Prime Numbers

Authors: Emmanuil Manousos
Comments: 5 Pages.

"The Octets of the Odd Numbers" theory categorizes the odd numbers into four categories D1, Q1, D2, Q2. We relate the distribution of octets of odd numbers in the set of integers to the distribution of prime numbers and obtain an algorithm for finding the set of prime numbers of the form D2 and Q2. The algorithm sequentially finds all prime numbers of the form D2 and Q2 in ascending order.
Category: Number Theory

P=NP? Proving the Problem

Authors: Mesut Kavak
Comments: 3 Pages.

The question about the problem is pretty clear:"Can any question whose solution can be quickly verified, also quickly solved?"
Category: Number Theory

Number System in Base One Hundred

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this paper I show my work on the possibility of make a number system in base one hundred, showing the table of possible assignations for every symbol and showing examples and conclusions.
Category: Number Theory

Analytical Proof of 3x + 1

Authors: Samuel Ferrer Colas
Comments: 5 Pages.

The Collatz or 3x + 1 conjecture is perhaps the simplest stated yet unsolved problem in mathematics in the last 70 years. It was circulated orally by Lothar Collatz at the International Congress of Mathematicians in Cambridge, Mass, in 1950 (Lagarias, 2010).The problem is known as the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem.In this concise paper I provide a proof of this conjecture, by finding an upper bound to the Collatz sequence and, as a consequence, a contradiction.
Category: Number Theory

Proof of the Riemann Hypothesis

Authors: Marcello Colozzo
Comments: 28 Pages.

We prove the Riemann Hypothesis by studying the behavior of a holomorphic function which has the same non-trivial zeros as the Riemann zeta function.
Category: Number Theory

The Semiprime Equivalent Proof of the Goldbach Conjecture

Authors: Stephen Marshall
Comments: 6 Pages.

In number theory, for very difficult Number theory problems that have been open and unsolved for long periods of time it can often be wise to take alternative approaches to the problem. There more old unsolved Number Theory problems than most would think. The Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics, it has been unsolved for over 281 years. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:Every even integer which is ≥ 4 can be written as the sum of two primes. It also states that every even natural number greater than 2 is the sum of two prime numbers. Or more specifically, that the "strong" Goldbach Conjecture asserts that all positive even integers ≥ 4 can be expressed as the sum of two primes. Two primes (p,q) such that p + q = 2n for n a positive integer ≥ 2.The conjecture has been shown via computer to hold for all integers less than 4×1018, but remains unproven despite enormous effort by many mathematicians over hundreds of years. Even the author has spent much effort attempting to solve this conjecture using several different direct methods and have come very close but was not able to prove the Goldbach Conjecture using any of these direct approaches. All of this effort made the author realize how difficult the Goldbach Conjecture is to solve using direct approaches, so this made him consider looking for a back door approach, or a work around the direct approaches. Any such approach could be different than the Goldbach Conjecture, but if it is a different Conjecture must be the equivalent of the Goldbach Conjecture, conjecture otherwise it would not solve the Goldbach Conjecture. This is exactly what the author has done, an equivalent conjecture has been developed and proven, thus solving the Goldbach Conjecture. Therefore, we call this a "back door" proof of the Goldbach Conjecture.
Category: Number Theory

Collatz Problemi İspatı (Proof of the Collatz Problem)

Authors: Mesut Kavak
Comments: 4 Pages. In Turkish

Pozitif bir tam sayı seçildiğinde, sayı çift ise 2'ye bölünür; aksi takdirde 3 ile çarpılır ve bundan sonra sonuca 1 eklenir. Sonucun tek veya çift olması şartından dolayı problemin gerekli seçeneği ile aynı işlem tekrarlandığında, 0 ve 1'den farklı olan her pozitif tamsayı 1'e indirgenebilir mi?

Choosing a positive integer is divisible by 2 if the number is even; otherwise it is multiplied by 3 and then 1 is added to the result. Can every positive integer other than 0 and 1 be reduced to 1 when the same operation is repeated with the required option of the problem, provided that the result is odd or even?
Category: Number Theory

İkiz Asallar Kestirimi İspatı (Proof for Twin Prime Conjecture)

Authors: Mesut Kavak
Comments: 3 Pages. in Turkish

İkiz asallar, aralarındaki fark 2 olan asal sayılardır. Sonsuz sayıda ikiz asal sayı var mıdır?Twin primes are prime numbers that differ by 2. Are there an infinite number of twin prime numbers?
Category: Number Theory

Proof for Twin Prime Conjecture

Authors: Mesut Kavak
Comments: 3 Pages.

"Twin primes are prime numbers that differ by 2. Are there infinitely many twin primes?"
Category: Number Theory

Proof for Collatz Conjecture

Authors: Mesut Kavak
Comments: 4 Pages.

When a positive integer is chosen, if the number is even, it is divided by 2; otherwise, it is multiplied by 3 and after that 1 is added to the result. Due to the condition that the result is odd or even, the same operation is repeated with the required option of the problem, every positive integer other than 0 and 1 can the integer be reduced to 1?
Category: Number Theory

Considerations on the 3n+1 Problem

Authors: V. Barbera
Comments: 9 Pages.

This paper presents some considerations on the 3n+1 problem. In particular on the next odd elements in the sequence lower than the starting number.
Category: Number Theory