Related papers: There are infinitely many cousin primes

The paper gives a unified and simple proof of both theorems and Cousin's theorem.

We prove several extensions of the Erdos-Fuchs theorem.

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means…

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.

It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite

We survey the classical results on the prime number theorem

I give some claims on primorial prime numbers for interested readers in number theory.

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order.…

The Schinzel hypothesis essentially claims that finitely many irreducible polynomials in one variable over Z simultaneously assume infinitely many prime values unless there is an obvious reason why this is impossible. We prove that under a…

Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…

A proof of Sendov's conjecture is given.

For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and…

In this paper, we prove that for any $A>0$ there exist infinitely many primes $p$ for which sums of the Legendre symbol modulo $p$ over an interval of length $(\ln p)^A$ can take large values.

We settle in the affirmative the Graham-Sloane conjecture.

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…

It is proved that any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.

The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the…

In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set,…

We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomass\'e. The main tools are the notion of well quasi ordering and the correspondence between cographs and some…

In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.