Number Theory

The Field of Q-Adic Numbers

Authors: Antoine Balan
Comments: 1 Page. Written in french

Here is defined a field like the p-adic numbers. The new field is called the field of q-adic numbers and depends only of $q=p^n$.
Category: Number Theory

On Chowla's Conjecture

Authors: Theophilus Agama
Comments: 5 Pages.

In this paper, using the area method, we establish Chowla's conjecture concerning the two point correlation of the Liouville function.
Category: Number Theory

About the Derivation of a Sequence of Real Numbers

Authors: Antoine Balan
Comments: 2 Pages. In french

Here is defined the derivation of a sequence of real numbers with help of a basis for the integers.
Category: Number Theory

Vii. la Conjetura De Collat: Orden Y Armonía en Los Números de Las Secuencias

Authors: Miguel Cerdá Bennassar
Comments: Pages.

Los nodos de color azul representan grupos de números pares, clasificados según el valor de su raíz digital y los de color rojo, grupos de números impares, clasificados del mismo modo. Llamamos a estos grupos rd(n), vértices con sus aristas que indican la dirección de las iteraciones. Todos los números impares de una secuencia de Collatz iteran a números pares de rd(1), rd(4) y rd(7). Los tres conjuntos representan las iteraciones de los números en una secuencia de Collatz, cuando de la función aplicada a un número par resulta un número impar y aplicada a un número impar, resulta un número par.
Category: Number Theory

The Super-Generalised Fermat Equation Pa^x + Qb^y=rc^z and Five Related Proofs

Authors: Julian Beauchamp
Comments: 11 Pages.

In this paper, we consider five proofs related to the super-generalised Fermat equation, Pa^x + Qb^y=Rc^z. All proofs depend on a new identity for a^x + b^y which can be expressed as a binomial sum to an indeterminate power, z. We begin with the Generalised Fermat Conjecture, for the case P,Q,R=1, also known as the Tijdeman-Zagier Conjecture and Beal Conjecture. We then show how the method applies to its famous corollary Fermat's Last Theorem, where x,y,z=n. We then return to the title equation, considered by Henri Darmon and Andrew Granville and extend the proof for the case P,Q,R>1 and x,y,z>2. Finally, we use the results to prove Catalan's Conjecture, and from this a weak proof that under certain conditions only one solution exists for equations of the form a^4-c^2=b^y.
Category: Number Theory

Indexed Compression

Authors: Arun Jose
Comments: 3 Pages.

This paper examines a lossless text compression algorithm that works on the principle that in any text file with recognizable words, the vast majority of character sequences would be recorded in a comprehensive dictionary, and can thus be mapped to bits directly, taking advantage of information entropy. The algorithm is tested on a sample book file, and compression factor and time are reported.
Category: Number Theory

Additive-Multiplicative Functions and the Reimann Zita Function

Authors: Mohammed Zohal
Comments: 7 Pages. please send your remarks to mohammedzohal2016@gmail.com

This article [studies] the sum ∑_d/n f(d), with f [being] an arithmetical function which not only multiplicative, but, of another form which will be additive_multiplicative, you will see what is it about, in fact, it has generated incredible formulas, which can give insight into the random arithmetic world of numbers.
Category: Number Theory

The Circle Embedding Method and Applications

Authors: Theophilus Agama, Berndt Gensel
Comments: 48 Pages. A partial proof of the Goldbach conjecture added.

In this paper we introduce and develop the circle embedding method. This method hinges essentially on a Combinatorial structure which we choose to call circles of partition. We provide applications in the context of problems relating to deciding on the feasibility of partitioning numbers into certain class of integers. In particular, our method allows us to partition any sufficiently large number $n\in\mathbb{N}$ into any set $\mathbb{H}$ with natural density greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on class of problems of similar flavour. The paper finishes by giving a partial proof of the binary Goldbach conjecture.
Category: Number Theory

A Sieve for Goldbach Conjecture

Authors: Xuan Zhong Ni
Comments: 3 Pages.

In this article, we find a new sieve for pair of primes whose summation equals to a given Even Number.
Category: Number Theory

A Classic Algebraic Identity Implies Fermat's Last Theorem (FLT) for Integral Exponent Larger Than Two

Authors: Philip A. Bloom
Comments: 2 Pages.

We solve the open problem of a simple proof of FLT for n > 2 by directly inferring, from Euclid's formula, a generalization that holds for the set of all coprime triples, a set that is equal to the set of all coprime triples {(z, y, x )} for which z ^ n - y ^ n = x ^ n holds. Our generalization allows us to deduce a necessary condition for coprime {(z, y, x)} to satisfy the Fermat equation z ^ n - y ^ n = x ^ n, the condition being that n is not larger than two.
Category: Number Theory

A New Sieve for Twin Primes

Authors: Xuan Zhong Ni
Comments: 2 Pages.

In this article, we find a new sieve for twin primes to prove the twin prime theory.
Category: Number Theory

Elementary Proof that the Goldbach Conjecture Is False

Authors: Stephen M. Marshall
Comments: 5 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. 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 (the strong conjecture) He then proposed a second conjecture in the margin of his letter: Every odd integer greater than 5 can be written as the sum of three primes (the weak conjecture). In number theory, Goldbach's weak conjecture, also known as the ternary Goldbach problem, states that every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott finally proved Goldbach's weak conjecture, a huge contribution to mathematics and number theory. The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history. The author would like to give many thanks to Harald Helfgott for his proof of the weak conjecture, because this elementary proof of showing the strong conjecture is false, was completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.
Category: Number Theory

Omni, A Mathematics Model of Our Spiritual Reality

Authors: Prateek Goel
Comments: 77 Pages. The author may be contacted at pratgoel@gmail.com.

Title: Omni, A Mathematics Model of Spiritual Reality
Author:  Kushal Prateek Goel

Abstract:
The experimental sciences declare that if it can’t be weighed or measured, then it isn’t scientific.

But where does that leave spirituality?  Spiritualists declare a truth to this existence that is before and beyond any measurable phenomena. 

What to do?  Do we pass off spirituality as nonsense because it proclaims a truth that cannot be observed under a microscope or verified through a telescope?  Or are we forced into a stalemate: all things spiritual can never be scientific and vice versa? 

This book presents a different possibility.  It presents the following positions:

1. There is a truth to this existence that is before and beyond material phenomena.
2. Mathematics is the only science that can possibly model that truth because Mathematics is the only science that does not require observable phenomena for its success. (The fact of 1 + 1 = 2 is based on constancy of definition for those symbols, not on 1 apple plus 1 apple equaling 2 apples.)
3. We can use simple Mathematics to classify aspects of this existence that are “spiritual”, that is, before and beyond material phenomenality.
4. First, we explore the mathematical properties of 0, 1 and Infinity and their application to this reality:

• We use 0 and its math properties to model consciousness or the ultimate ground of Being.
• We use 1 and its math properties to model the body, the identity, the Universe, and anything else that can be grasped by the hand, eye or mind.
• We use Infinity and its math properties to model the synapse and sentience.
• We use Infinity to model why we experience a subjective-objective split in this reality.

1. 0, 1 and Infinity (including arithmetic operations and the square root) are posited as the necessary and sufficient basis for all numerical values and all geometries. We posit these three are the fundamental generators of our reality too.
2. Not only are these 3 values foundational in Mathematics, they are spoken of by spiritualists across spiritual traditions. Take one example from Christianity: the Father (0), the Son (1) and the Holy Spirit (∞).  These concepts exist in virtually all spiritual traditions and are discussed at length.
3. We call this math-based spiritual model of existence “Omni” and give it a symbol for convenience and organizational purposes only. The name and symbol are utterly unimportant to the positions of the book.
4. We can use only the values 0, 1 and Infinity and the universal phenomenon of the cycle to mathematically derive a 4^th value: Phi, or the Golden Ratio.
5. We show how the fractal of the Golden Ratio is a model that yields at least 5 unique properties of our existence, properties that are spoken of in many spiritual traditions.
6. Having explored these, we look at some applications of the findings, for example a re-organization of spiritual practices that have been outlined in various spiritual traditions, or for example, the organization of spiritual community.

Infinite Twin Primes: Two Methods

Authors: William Blickos
Comments: 27 Pages. thoughtfarm@live.com

This paper is a revision and consolidation of two different but related methods to prove that there are infinitely many twin primes. The proofs are presented in the opposite order in which they were developed, largely due to the fact that a statement used at the end of the original proof, requiring its own proof, inspired and lead to the second method. The original technique uses surfaces, parabolas, and a number of lines. The 2nd proof, presented 1st, is actually the more direct and formal method. It primarily uses 2 surfaces, and also includes the extra steps needed to prove an analogous statement to one that was treated trivially in the first. Together these proofs compliment each other, and contain my body of work on the subject in a single resource.
Category: Number Theory

New Formula for Prime Counting Function

Authors: Majid azimi
Comments: 6 Pages.

This paper presents two functions for prime counting function and its inverse function (the function that returns nth prime number as output) with high accuracy and best approximation, which, due to their significant features, are distinguished from other similar functions presented thus far. the presented function for prime counting function is denoted by πm(x) and presented function for nth prime is denoted by Pm(x) in this article.
Category: Number Theory

Riemann Hypothesis and Value Distribution Theory

Authors: JinHua Fei
Comments: 7 Pages.

Riemann Hypothesis was posed by Riemann in early 50’s of the 19 th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper”problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the ‘critical line’. In this paper, we use Nevanlinna's Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis, in this paper, we are given out the second proof, please reader reference.
Category: Number Theory

Proof of Goldbach's Conjecture

Authors: Jaejin Lim
Comments: 2 Pages.

I prove Goldbach’s conjecture, ‘Every even integer greater than 2 can be expressed as the sum of two primes.’. And I used “Generalization of mathematical induction”.
Category: Number Theory