Number Theory

The Collatz Conjecture - A Definitive Proof

Authors: Peter G. Bass
Comments: [Pages resized by viXra Admin.]

This paper presents a proof of the Collatz Conjecture by showing that the sequence of numbers generated cannot either diverge, converge to a single number, possess more than one stable oscillation, or alternate indefinitely. The methods used are (i) Number pattern Recognition, (ii) basic analysis and (iii) simple interpretive logic.
Category: Number Theory

The Goldbach Conjecture - A Definitive Proof

Authors: Peter G. Bass
Comments: 4 Pages.

This paper presents a proof of the Goldbach Conjecture by comparing the distribution of prime numbers with the inverse distribution of odd composite numbers.
Category: Number Theory

An Inequality Approach of the Collatz Conjecture

Authors: Adarsha Chandra
Comments: 17 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

We define the Collatz Function F_col:N->N(n) as follows: F_col(n):= n/2 if n is even, and F_col(n):= 3n+1 if n is odd We define the two branches f:N->N and g:N->N of the above function as follows: f(n):= n/2 if n is even and g(n):= 3n+1 if n is odd Also, we define the 'functional sequence' of a number n as the set of functions applied consecutively on n (obeying the obtained parities), and show that any two g's in a functional sequence must be separated by at least one f. Next, we prove that all numbers n, under repetitive execution of the Collatz function, eventually yield a certain E < n. This is obvious for even n values, since- F_col(n) = n/2 < n for even n For odd n values, we prove that any odd number n which does not yield an E < n under repetitive execution of the Collatz function, must possess a functional sequence of the form- S={gfgfgfgfgfgf...} We then prove that the existence of a number possessing such a functional sequence is not possible, implying that our statement is true for odd numbers as well. Hence, it follows that any natural number n, under repetitive execution of the Collatz function, must yield an E < n. The truth of the Collatz Conjecture follows immediately from the above.
Category: Number Theory

Equivalent ABC Conjecture Proved on Two Pages

Authors: A. A. Frempong
Comments: 6 Pages. Copyright © by A. A. Frempong

By applying basic mathematical principles, the author proves an equivalent ABC conjecture.The equivalent ABC conjecture proved in this paper states that for every positive real number,ε, there exists only finitely many triples (A, B, C) of coprime positive integers, with A + B = C, such that C<K(rad(d))^(1+ε),where d is the product of the distinct prime factors of A, B and C; and K is a constant. From the hypothesis, A + B = C, it was proved that C<K(rad(d))^(1+ε).
Category: Number Theory

Notes on the Prouhet–Tarry–Escott Problem

Authors: Philip Gibbs
Comments: 12 Pages.

A summary of selected solutions to the ideal Prouhet–Tarry–Escott problem up to size 7
Category: Number Theory

Proof of the Riemann Hypothesis

Authors: Guilherme Rocha de Rezende
Comments: 6 Pages.

In this article we will prove the problem equivalent to the Riemann Hypothesis developed by Luis-Báez in the article ``A sequential Riesz-like criterion for the Riemann hypothesis''.
Category: Number Theory

Proof of the Generalized Riemann Hypothesis

Authors: Jayesh Mewada
Comments: 36 Pages.

A beautiful pattern in Dirichlet Eta function is found. A new mathematical operator is introduced which reduces Dirichlet Eta function into a new function. It is shown that the function allows for 'Zero' at only one value of real component within the critical strip for any given value of imaginary component. Consequently it can be concluded that non-trivial 'Zeros' of Riemann Zeta function can only be on critical line, thereby proving the Riemann Hypothesis with absolute certainty. It is also shown that all the non-trivial 'Zeros' of the Riemann Zeta are simple 'Zeros'. It is then shown that the same method of proof can be generalised to the other Dirichlet L-Functions with suitable modifications, thereby proving the Generalised Riemann Hypothesis also to be true
Category: Number Theory

Is The Riemann Hypothesis True? Yes, It Is. v(4)

Authors: Abdelmajid Ben Hadj Salem
Comments: 9 Pages. Submitted to the journal " Communications On Pure and Applied Mathematics. Comments welcome.

In 1859, Georg Friedrich Bernhard Riemann announced the following conjecture, called Riemann Hypothesis: The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by: $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$have real part $sigma= ds frac{1}{2}$.We give the proof that $sigma= frac{1}{2}$ using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet $eta$ function.
Category: Number Theory

Proof of the Strong Goldbach Conjecture

Authors: Kurmet Sultan
Comments: 8 Pages.

The article provides a proof of the Strong (binary) Goldbach conjecture based on the regularities of the interval between successive primes.
Category: Number Theory

The Introduction of the Rule of 3x +1 Problem

Authors: Li Ke
Comments: 7 Pages. Innovative ideas and methods.

This article introduces a change rule of 3x+1 problem (Collatz conjecture), it’s named LiKe’s rule: For any positive integer, change by the Collatz conjecture, it will change to an odd number; the odd numbers will must change to a number of LiKe’s second sequence {3n-1∣n∈Z+}; then this 3n-1 will change to a smaller 3n-1 and gradually decrease to 8(that is 3^2-1) then back to 1 in theend.
Category: Number Theory

Diffusion Approximation for Counting Number of Primes

Authors: Gregory M. Sobko
Comments: 15 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Quite deterministic nature of prime numbers, due to the complexity of the recurrent generating algorithms, is mimicking ‘randomness’ and stimulates to apply probabilistic instruments to analyze number-theoretic problems. The key issue in the probabilistic analysis in a number-theoretic framework remains an enigmatic connection between the deterministic nature of integer sequences related to prime numbers and their apparent complicated (‘unpredictable’ or ‘chaotic’) behavior interpreted as ‘randomness’. We derive multiplicative and additive models with recurrent equations for generating sequences of prime numbers based on the reduced Sieve of Eratosthenes Algorithm and analyze their asymptotic behavior with the help of Riemann Zeta probability distribution. This allows interpreting such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by prime numbers, representing paths of stochastic dynamical systems. We analyze in this work an additive continuous-time probabilistic model of counting function of primes pi(n) in terms of diffusion approximation of non-Markov random walks. We assume that ‘updating’ terms eta in the recurrent equation pi(n(k+1)) - pi(n(k)) = eta(n(k+1) follow Zeta probability distribution and calculate infinitesimal characteristics of the random walk, which approximate coefficients of the corresponding stochastic differential equation. Computer modeling illustrates graphically an impressive fitting of trajectories for the original counting function, the calculated trend function, and the Brownian approximation.
Category: Number Theory

Restrictions to Goldbach’s Conjecture Using Bertrand’s Postulate, Initiation to a Proof

Authors: Anass Massoudi
Comments: 8 Pages. [Fonts reduced by viXra Admin]

The Goldbach conjecture, also named the binary Goldbach’s conjecture, proposed by the Russian mathematician Christian Goldbach in 1742, states that for every even integer n bigger than 2, there is always two primes a and b such that n = a + b, and until now this conjecture remained unproven, in this paper, we use what’s known as Bertrand’s postulate to restrict the conditions for the two primes a and b that verify this conjecture for every even number n = 2p, namely proving the interesting fact that the Goldbach statement is valid if and only if we have p < a < 2p – 2 and b < p, leading to a clue to prove this conjecture in a simpler manner than attack it brutally without any knowledge about the properties of a and b and the inequality that we will prove, making at last an initiation for a proof of the Goldbach statement .
Category: Number Theory

Diophantine Quadruples and Ideal Solutions of the Prouhet–Tarry–Escott Problem of Size Four

Authors: Philip Gibbs
Comments: 8 Pages.

A rational Diophantine m-tuple is a set of m rational numbers such that the product of any two is one less than a square. The Prouhet-Tarry-Escott problem seeks two different multisets of n integers such that the sums of like powers of each set are equal for all exponents up to some k < n. Here a new connection is established between rational Diophantine quadruples (m=4) and ideal solutions of the Prouhet–Tarry–Escott problem of size 4 (n=4, k=3) Both problems are shown to be related to finding 3 by 3 singular matrices of integers whose 9 elements are all square.
Category: Number Theory

Two Proofs of Riemann Hypothesis by Vector Properties of Riemann Zeta Function

Authors: Tae Beom Lee
Comments: 24 Pages.

The Riemann zeta function(RZF) ζ(s) is useful in number theory for studing properties of prime numbers. The Dirichlet eta function(DEF) η(s) is modification of RZF. In this thesis, we treat each term of RZF and DEF as a vector. From the geometric properties of vectors, we got clues of proof from the fact that, in a complex variable s = α + iβ, α only affects the magnitude of each vector and β affects only the argument of each vector, independently. So, each vector with same n are parallel to each other, regardless of the value of α. This parallel property implies a very strict geometric restriction which lead to two successful proofs of Riemann Hypothesis(RH). One proof is from the contradictions which come from the trajectories of RZF, and the other proof is by applying Chauchy integral theorem to the trajectory of RZF. We tried to provide sufficient graphs and videos for the understanding of the vector geometry properties of RZF and DEF. In appendix, we provided the source programs for analyzing vectors and suggested two other possible proofs of RH for further studies.
Category: Number Theory

Asymptotic Distribution of Residuals Within Congruence Classes Generated by Primes

Authors: Gregory M. Sobko
Comments: 10 Pages.

By using the Dirichlet characters for a finite abelian group Gp = Zp=Z/(pZ), where p is a prime, and the corresponding characteristic functions Ф(v(n)), we discuss asymptotic distribution for sums of residuals r = mod(v, p) = [v]p for p from P, a set of all prime numbers. Here vi is a random variable with a certain probability distribution on the set of all natural numbers N, and v(n) is a sum v1 + v2 + … + vn of independent random integers (not necessarily equally distributed). We prove that the residuals of sums [v(n)]p = [v1]p +[v2] + … + [vn]p are asymptotically uniformly distributed on Gp for every prime p ( Gp is a congruence class generated by p ). Then, we prove that the components of the vector of residuals r(v(n))=(r1, r2, …, rpi(n)) are asymptotically independent random variables.Type equation here.
Category: Number Theory

Beyond A Quartic Polynomial Modeling the DNA Double Helix Genetic Code

Authors: Hans Hermann Otto
Comments: 14 Pages.

By combination of finite number theory and quantum information the complete quantum information in the DNA genetic code has been made likely by Planat et al. (2020). In the present contribution a varied quartic polynomial contrasting the polynomial used by Planat et al. is proposed that considered apart from the golden mean also the fifth power of this dominant number of nature to adapt the code information. The suggested polynomial is denoted as 〖g(x)=x〗^4-x^3-(4〖-φ^2)x〗^2+(4-φ^2)x+1, where φ=(√5-1)/2 is the golden mean. Its roots are changed to more golden mean based ones in comparison to the Planat polynomial. The new coefficients 4-φ^2 instead of 4 would implement the fifth power of the golden mean indirectly applying 4-φ^2=3+φ=√(13+φ^5 )=√(2+φ^(-5) )=3.6180... As an outlook it should be emphesized that the connection between genetic code and resonance code of the DNA may lead us to a full understanding of how nature stores and processes compacted information and what indeed is consciousness linking everything with each other suggestedly mediated by all-pervasive dark constituents of matter respectively energy.
Category: Number Theory

Using the Partial Sums of the Alternating Harmonic Series to Prove the Harmonic Series is Divergent

Authors: Robert Spoljaric
Comments: 5 Pages.

Many proofs of the divergence of the harmonic series have been given since the first proof by Nicole Oresme (1323-1382). In this article we shall give a simple proof using the partial sums of the alternating harmonic series. A simple consequence of this is an approximation that follows as a corollary. We then show that every harmonic number is the sum of partial sums of the alternating harmonic series. Finally as a corollary we show that the sequence of subseries of the harmonic series is converging to ln2.
Category: Number Theory

A Fresh Hope of Proving Goldbach Conjecture (1+1)

Authors: Li Ke
Comments: 3 Pages. -

This paper provide a new way of proving Goldbach conjecture - LiKe sequence (This method was published in 2019). And briefly introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any prime sequence (3,5,7,…,Pn), there must have no LiKe sequence less than 3×Pn. This method only studies prime numbers and composite numbers, which is very important for the study of Goldbach conjecture.
Category: Number Theory

New Argentest Primality Test Algorithm

Authors: Zeolla Gabriel Martin
Comments: 23 Pages. Original language Spanish translated into English

This text develops a new Primality Algorithm, this one obtains opposite results to Fermat's little theorem, since it uses similar mechanisms but applied to the analysis of patterns. In Fermat's Theorem there are always Pseudoprimes hidden among the primes, which does not give certainty about the primality of an odd number analyzed, beyond the change of bases as happens with the Pseudoprime number 561. In the Argentest algorithm, the opposite happens, the pseudoprimes do not pass the test, so we can confirm the primality of a number with absolute certainty and determination, but there is a percentage of primes that do not pass the test either, so we go to the change of base to re-analyze the patterns and confirm primality later.
Category: Number Theory

Proof of Infinity of Twin Primes and Cousin Primes

Authors: Kurmet Sultan
Comments: 3 Pages. In Russian

The article provides a proof of the infinity of twin primes and cousin primes, as well as a proof of the correctness of the Polignac conjecture.
Category: Number Theory

Function of Two Variables and Sequences Eventually Periodic

Authors: Miquel Cerdà Bennassar
Comments: 5 Pages.

I present an algorithm that defines a function generator of sequences eventually periodic, with eligible cycle values and starting with any integer.
Category: Number Theory

Proof of Polignac's Conjecture for Pap Equal to Eight

Authors: Marko V. Jankovic
Comments: 17 Pages.

In this paper proof of the Polignac's Conjecture for gap equal to eight is going to be presented. It will be shown that consecutive primes with gap eight could be obtained through two stage sieve process, and that will be used to prove that infinitely many primes with gap eight exist. The proof represents an simple extension of the recently presented proof that infinitely many sexy prime exist. The major contribution of this paper is presentation of all elementary modules that are necessary for the proof of Polgnac's conjecture in general case.
Category: Number Theory

In Situ Experiment on Fractal Corresponds with Cosmological Observations and Conjectures

Authors: Blair D. Macdonald
Comments: 30 Pages.

Fractal geometry is an accepted mathematical description of nature. One of the great questions in cosmology—along with what is the ‘dark energy’ and the other cosmic anomalies—is whether the universe is also fractal? The 2012 WiggleZ Dark Energy Survey found, in agreement with fractal-cosmology proponents, the small-scale observable universe is fractal, the large-scale is not fractal. Fractals have not been modelled from the perspective of being within a growing one. Can a (different) fractal model explain all cosmological observations and conjectures, and if so, are we are modelling the fractal universe incorrectly? An experiment was conducted on a ‘simple’ (Koch snowflake) fractal, testing the perspective of an in-situ observer within a growing/emergent fractal — ‘looking back’ in iteration-time to its origin. New triangle sizes were held constant allowing earlier triangles in the set to expand as the set iterated. Classical kinematic equations of velocities and accelerations were calculated for the total area total and the distance between points. Hubble-Lemaitre's Law and other cosmological observations and conjectures were tested for. Results showed area(s) expanded exponentially from an arbitrary starting position; and as a consequence, the distances between points — from any location within the set — receded away from the ‘observer’ at increasing velocities and accelerations. It was concluded, at the expense of the cosmological principle, that the fractal is a geometrical match to the cosmological problems, including the inflation epoch, Hubble-Lemaitre and accelerated expansion; inhomogeneous (fractal) galaxy distribution on the small and homogenous on large scales; and other problems — including the cosmological catastrophe. The fractal may offer a direct mechanism to the cosmological problem and can further explain the quantum problem — unifying the two realities as being two aspects of the same geometry.
Category: Number Theory

Nuevo Algoritmo de Prueba de Primalidad Argentest (New Argentest Primality Test Algorithm)

Authors: Zeolla Gabriel Martin
Comments: 23 Pages. Español

Este texto desarrolla un nuevo Algoritmo de Primalidad, este obtiene resultados opuestos al pequeño teorema de Fermat, ya que utiliza mecanismos similares pero aplicados al análisis de patrones. En el Teorema de Fermat siempre hay Pseudoprimos escondidos entre los primos, lo cual no da certezas sobre la primalidad de un número impar analizado, más allá del cambio de bases como sucede con el número Pseudoprimo 561. En el algoritmo Argentest sucede lo contrario los pseudoprimos no pasan el test, por lo cual podemos confirmar la primalidad de un número con absoluta certeza y determinación, pero hay un porcentaje de primos que tampoco pasan el test, por lo cual acudimos al cambio de base para volver a analizar los patrones y confirmar la primalidad luego.

This text develops a new Primality Algorithm, this one obtains opposite results to Fermat's little theorem, since it uses similar mechanisms but applied to the analysis of patterns. In Fermat's Theorem there are always Pseudoprimes hidden among the primes, which does not give certainty about the primality of an odd number analyzed, beyond the change of bases as happens with the Pseudoprime number 561. In the Argentest algorithm, the opposite happens, the pseudoprimes do not pass the test, so we can confirm the primality of a number with absolute certainty and determination, but there is a percentage of primes that do not pass the test either, so we go to the base change to re-analyze the patterns and confirm primality later
Category: Number Theory