Related papers: Consecutive primes and IP sets

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…

In this paper, we provide versions of Van der Waerden's theorem and Rado's theorem for finite colorings of IP-sets and k-IP-sets. Here, by an IP-set we mean a set of integers that contains all finite sums of an infinite subset of N, and we…

In this paper we work with number sequences from the On-Line Encyclopedia of Integer Sequences (OEIS). Using the Pepis-Kalmar pairing function, we obtain an infinite matrix of natural numbers in which odd natural numbers are separated from…

For every prime $p$, we construct an infinite countable group that contains precisely $p-1$ elements which are not $p$th powers.

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical…

We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five…

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively…

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has…

By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a…

If $p$ is prime, a sequence of prime numbers $\{p, 2p+1, 4p+3,...,2^{n-1}(p+1)-1\}$ is called a Cunningham chain. These are finite sequences of prime numbers, for which each element but the last is a Sophie Germain prime. It is conjectured…

The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in…

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set…

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability…

A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[…

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\leq x$ such that a non-empty set of irreducible polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this set can…

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime…

Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with…

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…