Related papers: On a Weakened Form of Polignac Conjecture

For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any…

An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $\pi(kn)$ is prime, where $\pi(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive…

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and…

Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha$ is a $p$-adic integer with $\alpha\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that…

The world of primes has many gaps between evidence and theorems. Here, we review Legendre's conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming…

We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic…

Let $\epsilon\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1…

Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively.…

Two centuries ago, Sophie Germain began to work on her grand plan to prove the theorem of Fermat, the famous conjecture that $x^n + y^n = z^n$ is impossible for nonzero integral values of $x$, $y$, and $z$, when $n > 2$. At that time, this…

We study the set $S(q)$ of residue classes $r$ modulo the Pisano period $\pi(q)$ for which $q \mid \varphi(F_m)$ for every $m \equiv r \pmod{\pi(q)}$. We prove that if $q$ is a Sophie Germain prime and $z(2q+1) \mid \pi(q)$, then $S(q)$ is…

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists…

Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every…

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

Let $m$ be a Carmichael number and let $L$ be the least common multiple of $p-1$, where $p$ runs over the prime factors of $m$. We determine all the Carmichael numbers $m$ with a Fermat prime factor such that $L=2^{\alpha}P^2$, where $k\in…

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…