Number Theory

Fermat and Mersenne Prime Criteria for the Infinity or the Strong Finiteness of Primes of the Form 2^x±1

Authors: Pingyuan Zhou
Comments: 29 Pages. Author pesents a mathematical partten in which Fermat and Mersenne primes can become criteria for the infinity or the strong finiteness each other.

Abstract: This paper presents that Fermat primes and Mersenne primes can separately become criteria for the infinity or the strong finiteness of primes of the form 2^x±1, which includes Fermat prime criteria for the set of Mersenne primes and its two subsets as well as Mersenne prime criteria for the set of Fermat primes and its two subsets.
Category: Number Theory

Fermat Prime Criterion Related to Landau's Fourth Problem

Authors: Pingyuan Zhou
Comments: 9 Pages. Anthor gives an argument for solving Landau's fourth problem from a hypothesis on the infinity of primes represented by quadratic polynomial of Mersenne primes.

Abstract: In this paper we consider the infinity of primes represented by quadratic polynomial with 4(Mp-2)^2+1, basing on a hypothesis as sufficient condition in which Fermat primes are criterion for the infinity of such primes, where Mp is Mersenne prime, and give an elementary argument for existence of infinitely many primes of the form x^2+1. As an addition, an elementary argument on the infinity of Mersenne primes is also given.
Category: Number Theory

Primality Tests for Specific Classes of Proth Numbers

Authors: Predrag Terzic
Comments: 3 Pages.

Polynomial time primality tests for specific classes of Proth numbers are introduced .
Category: Number Theory

The Sum of the Digits of a Number/primality Testing

Authors: Ounas Meriam
Comments: 04 Pages.

In this paper, I will try to explain my idea about the world of the digits of numbers which is somewhat circumvented by mathematicians.
Category: Number Theory

Perfect Cuboid Does not Exist

Authors: Saenko V.I.
Comments: Pages. The proof should be improved because in the present form it is valid only if all non-unit G_i are prime.

A perfect cuboid, i.e., a rectangular parallelepiped having integer edges, integer face diagonals, and integer space diagonal, is proved to be non-existing.
Category: Number Theory

Compositeness Tests for Specific Classes of K3^n+2

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness tests for specific classes of numbers of the form k3^n+2 are introduced .
Category: Number Theory

Compositeness Tests for Specific Classes of K3^n-2

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness tests for specific classes of numbers of the form k3^n-2 are introduced .
Category: Number Theory

Compositeness Test for Repunits Base 3

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness test for numbers of the form (3^p-1)/2 is introduced .
Category: Number Theory

A Proof of the Collatz Conjecture (After Second Modification)

Authors: Zhang Tianshu
Comments: 18 Pages. This is third manuscript for the article.

If every positive integer is able to be operated to 1 by the set operational rule of the Collatz conjecture, then begin with 1, we can get all positive integers by operations on the contrary of the set operational rule for infinite many times. In this article, we will apply the mathematical induction with the help of certain operations by each other’s- opposed operational rules to prove that the Collatz conjecture is tenable.
Category: Number Theory

A Proof of the Beal's Conjecture (After Second Modification)

Authors: Zhang Tianshu
Comments: 24 Pages. This is third manuscript for the article.

In this article, we first have proven a lemma of EP+FV≠2M. Successively have proven the Beal’s conjecture by mathematical analyses with the aid of the lemma, such that enable the Beal’s conjecture holds water.
Category: Number Theory

Conjectured Compositeness Tests for Specific Classes of B^n-B+1 and B^n+b-1

Authors: Predrag Terzic
Comments: 2 Pages.

Compositeness criteria for specific classes of numbers of the form b^n-b+1 and b^n+b-1 are introduced .
Category: Number Theory

When π(N) Does not Divide N

Authors: Germán Paz
Comments: 10 Pages. Some results and a question added.

Let $\pi(n)$ denote the prime-counting function and let
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$$f(n)=\left|\left\lfloor\log n-\lfloor\log n\rfloor-0.1\right\rfloor\right|\left\lfloor\frac{\left\lfloor n/\lfloor\log n-1\rfloor\right\rfloor\lfloor\log n-1\rfloor}{n}\right\rfloor\text{.}$$
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In this paper we prove that if $n$ is an integer $\ge 60184$ and $f(n)=0$, then $\pi(n)$ does not divide $n$. We also show that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $f(n)=1$. In addition, we prove that if $n\ge 60184$ and $n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\log n-1\rfloor$ located in the interval $[e^{\lfloor\log n-1\rfloor+1},e^{\lfloor\log n-1\rfloor+1.1}]$. This allows us to show that if $c$ is any fixed integer $\ge 12$, then in the interval $[e^c,e^{c+0.1}]$ there is always an integer $n$ such that $\pi(n)$ divides $n$.

Let $S$ denote the sequence of integers generated by the function $d(n)=n/\pi(n)$ (where $n\in\mathbb{Z}$ and $n>1$) and let $S_k$ denote the $k$th term of sequence $S$. Here we ask the question whether there are infinitely many positive integers $k$ such that $S_k=S_{k+1}$.
Category: Number Theory

Two Hundred and Thirteen Conjectures on Primes

Authors: Marius Coman
Comments: 145 Pages. Published by Education Publishing, USA. Copyright 2015 by Marius Coman.

This book brings together sixty-two articles regarding primes submitted by the author to the preprint scientific database Vixra, papers on squares of primes, semiprimes, twin primes, sequences of primes, ways to write primes, special classes of composites, formulas for generating large primes, formulas for generating different duplets and triplets of primes, generalizations of the twin primes and de Polignac’s conjectures, generalizations of Cunningham chains, Smarandache generalized Fermat numbers, and on many other issues very much related with the study of primes. Finally, in the last eight from these collected papers, I defined a new function, the MC function, and showed some of its possible applications: for instance, I conjectured that for any pair of twin primes (p, p + 2) there exist a positive integer n of the form 15 + 18*k such that the value of Smarandache function for n is equal to p and the value of MC function for n is equal to p+2 and I also made a Diophantine analysis of few Smarandache types sequences using the MC function.
Category: Number Theory

An Interesting Relation Between the Squares of Primes and the Number 96 and Two Conjectures

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make two conjectures based on the observation of an interesting relation between the squares of primes and the number 96.
Category: Number Theory

A Formula that Seems to Generate Easily Big Numbers that Are Primes or Products of Very Few Primes

Authors: Marius Coman
Comments: 2 Pages.

The formula N = (p^4 – 2*p^2 + m)/(m – 1), where p is an odd prime and m is a positive integer greater than 1, seems to generate easily primes or products of very few primes.
Category: Number Theory

Four Conjectures Based on the Observation of a Type of Recurrent Sequences Involving Semiprimes

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make four conjectures starting from the observation of the following recurrent relations: (((p*q – p)*2 – p)*2 – p)...), respectively (((p*q – q)*2 – q)*2 – q)...), where p, q are distinct odd primes.
Category: Number Theory

Statements on the Infinity of Few Sequences or Types of Duplets or Triplets of Primes

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make few statements on the infinity of few sequences or types of duplets and triplets of primes which, though could appear heterogenous, are all based on the observation of the prime factors of absolute Fermat pseudoprimes, Carmichael numbers, or of relative Fermat pseudoprimes to base two, Poulet numbers.
Category: Number Theory

The Proof for Non-existence of Perfect Cuboid

Authors: Bambore Dawit
Comments: 9 Pages. the proof is short cut, there are instructions and results

This paper shows the non-existence of perfect cuboid by using two tools, the first is representing Pythagoras triplets by two numbers and the second is realizing the impossibility of two similar equations for the same problem at the same time in different ways and the variables of one is relatively less than the other. When we express all Pythagoras triplets in perfect cuboid problem and rearrange it we can get a single equation that can express perfect cuboid. Unfortunately perfect cuboid has more than two similar equations that can express it and contradict one another.
Category: Number Theory

Compositeness Test for Repunit Numbers

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness test for numbers of the form (10^n-1)/9 is introduced .
Category: Number Theory

无穷大的运算法则

Authors: Liu Ran
Comments: 1 Page.

传统数论中的无穷大是没有上界的，也就是没有最大，只有更大。无穷大是自相矛盾的。
Category: Number Theory