Number Theory

The Proof [of] Goldbach’s Conjecture

Authors: Aniket Bhattacharjee
Comments: 2 Pages.

In this paper, I want to present the proof to one of the most famous conjecture - The Goldbach’s Conjecture.
Category: Number Theory

Sequences of Prime Numbers

Authors: Emmanuil Manousos
Comments: 6 Pages.

The categorization of odd numbers by "The octets of the odd numbers" theory gives an algorithm for finding prime numbers.
Category: Number Theory

Voyage on Two Syracuse

Authors: Berkouk Mohamed
Comments: 27 Pages. In French (Correction made by viXra Admin - Please conform!)

Dans une matrice carrée, vu qu’il y a deux suites, une horizontale de Syracuse (S0 ,S1,u2026Sm)Constituant les lignes de la matrice, puis des suites des valeurs prises de 0 à n pour un S0 donné, ces valeurs s’organisant à leurs tours verticalement dans les colonnes.le fait qu’elles soit déterminées respectivement par deux formules récurrentes issues des mêmes instructions de Collatz , méritent le nom de « deux Syracuse »

In a square matrix, since there are two sequences, a Syracuse horizontal (S0 ,S1,u2026Sm)Constituting the rows of the matrix, then the series of values u200bu200btaken from 0 to n for a given S0, these values u200bu200bbeing organized in turn vertically in the columns. the fact that they are respectively determined by two recurring formulas from the same instructions of Collatz, deserve the name of "two Syracuse"
Category: Number Theory

Proof of the Collatz Conjecture

Authors: Henok Tadesse
Comments: 29 Pages.

Take any positive integer N. If it is odd, multiply it by three and add one. If it is even, divide it by two. Repeatedly do the same operations to the results, forming a sequence. It is found that, whatever the initial number we choose, the sequence will eventually descend and reach number 1, where it enters an eternal closed loop of 1- 4 - 2 - 1. This has been numerically confirmed for initial numbers up to 260. This is known as the Collatz conjecture which states that the sequence always converges to 1. So far no proof has ever been found that this holds for every positive integer. This problem has been stated by some as perhaps the simplest math problem to state, yet perhaps the most difficult to solve. In this paper, we present a proof that the sequence always converges to 1.
Category: Number Theory

On Prime Number Generating Formula

Authors: Yuji Masuda
Comments: 1 Page.

Although there have been many attempts to obtain prime number generating formulas, the purpose of this study was to deepen our basic knowledge of prime numbers in order to create this prime number generating formula. I hope that this research will help to deepen our understanding of prime numbers.
Category: Number Theory

Solution to the Collatz Conjecture

Authors: Samuel Ferrer Colas
Comments: 5 Pages. The chart is only a hints. Do not consider it as part of the core proof

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

Proofs of ABC Conjecture

Authors: Dmitri Martila
Comments: 4 Pages. (Corrections made by viXra Admin to conform with scholarly norm)

Several crucial properties of ABC conjecture are presented and proven. Therefore, the ABC conjecture is proven.
Category: Number Theory

The Graphical Method of the Kakutani’s Problem

Authors: Xingyuan Zhang
Comments: 10 pages. This is my second proof of the Collatz conjecture called graphical method. It can be only 6 pages. It has also other solutions or ideas, such as only using a table and in binary. My first proof is at viXra: 2301.0154.

In this paper we had given another elementary proof of the Kakutani’s problem by using the Kakutani’s Angle, it holds. By detailed analysis of the properties of both forward and inverse operations of the proposition, we had some important conclusions: 1, there hasn’t any triple in the forward path numbers; 2, there have an infinity number of inverse path numbers which had been defined as similar numbers in one time of inverse operation; 3, on the figure of Kakutani’s Angle, the operation path of any odd is unique; 4, the inverse operations can start with any odd, and all of the path numbers on the countless paths is getting larger and larger, on the contrary, to do forward operations for any inverse path number, it must go back to the starting point or to 1. It’s not difficult to prove if understanding its operational mechanism and grasping the methods.
Category: Number Theory

Pythagorean Triples and the Binomial Formula

Authors: Kurmet Sultan
Comments: 3 Pages.

The article shows the possibility of compiling Pythagorean triples using the binomial formula and provides a Theorem that is an alternative proof of the infinity of Pythagorean triples and confirmation of the close connection of the Pythagorean Theorem with the binomial formula.
Category: Number Theory

Disproof of the Riemann Hypothesis and the Non-Trivial Zeros of the Zeta Function

Authors: Henok Tadesse
Comments: 7 Pages.

This paper disproves the Riemann hypothesis by disproving the non-trivial zeros of the Riemann zeta function.
Category: Number Theory

A proof of Riemann Hypothesis.

Authors: Liu Yajun
Comments: 2 Pages.

we give a proof of Riemann Hypothesis.
Category: Number Theory

An Elementary Proof of Collatz's Conjecture

Authors: Chongxi Yu
Comments: 3 Pages.

Any even or odd number can be written as one of 10x + 1, 10x + 3, 10x + 5, 10x + 7, 10x + 9, 10x + 0, 10x + 2, 10x + 4, 10x + 6, or 10x + 8, (x= 1, 2, 3,u2026.n); all 10x + 1, 10x + 3, 10x + 5, 10x + 7, 10x + 9, 10x + 0, 10x + 2, 10x + 4, 10x + 6, or 10x + 8, can be transferred in to 5 x 2y, y = 1, 2, 3, 4, 5, 6, 8,u2026m by repeating two arithmetic operation (3x + 1 and dividing 2). When y is an odd number, 3 times y plus 1 will always yield one even number, if the even number is not one of 2n, then the even number divide 2 once or more, a new odd number y’ will be yielded, but the new odd number must be different from the original y, 3 times y’ + 1 will yield another new even number, if the new even number is not one of 2n, then the new even number divide 2 once or more, a new odd number y’’ will be yielded, so on, every dividing operation will yield one new odd number which is different from previous odd number, every time 3 + 1 will yield a new even number which is different from previous even number, these operations can be going unlimited and infinite different even numbers will be yielded until reach one of 2n which is less than total even number, but is also infinite, that is: by an infinite number of repeating two arithmetic operation (3x + 1 and dividing 2), one of 2n must be reach, then 5 x 1 will be reach, final 1 will be reach, this statement must be true, then the Collatz’s conjecture will be the Collatz’s theorem
Category: Number Theory

Syracuse Conjecture Quadrature

Authors: Rolando Zucchini
Comments: 41 Pages.

After circa 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 — 212 BC) the history of mathematics repeats itself in a different problem.The conjecture of Syracuse, or Collatz conjecture, is approached from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a particular type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.
Category: Number Theory

The Symmetry of N-domain and Prime Number Conjectures

Authors: Yajun Liu
Comments: 2 Pages. (Author name added to article by viXra Admin - Please conform!)

In this paper, we discuss the symmetry of N-domain and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture and Twins Prime Conjecture.
Category: Number Theory

L1/2(0 1/2 1) Space and Quantum Time-Space with Energy

Authors: Yajun Liu
Comments: 5 Pages. (Note by viXra Admin: Author's name format should be first name followed by last name)

In this paper, We constructed a Time-Space with energy model just considering the velocity of the light C and the Plank constant h and "1/" a_g (a_g is the strength of gravition (m/s2)) This model has a geometry space (complex) and just provide a probability to combine the Gravitation and Electric-Magnetics field under a basic structure of quantum Time-Space with energy. We hope to throw a little bit light on the big picture of uniting the quantum mechanics and General relative theory.KeywordsQuantum Time-Space with energy Unified Field Theory
Category: Number Theory

Solution Conditions

Authors: Hajime Mashima
Comments: 38 Pages.

For Fermat’s Last Theorem, the condition that holds when there isinverse element.
Category: Number Theory