Related papers: Consecutive primes and IP sets

Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is…

The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…

In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal A,\mu,T)$, any…

Natural numbers satisfying an unusual property are mentioned by the author in [5], in which their infinitude is also proved. In this paper, we start with an arbitrary natural number which is not a multiple of 10 and non-palindromic, form…

For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…

Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of…

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a…

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 prove that it is always possible to find a permutation $p$ on the set $\{1,...,n\}$ such that $c+p(c)$ is prime for all $c \in \{1,...,n\}.$

In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p - 1)(q -…

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \ge 11$ and integer $r\ge 2$, we prove that $$ \sum\limits_{\begin{smallmatrix}…

Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of…

The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings,…

Let $E\subset \mathbb Z$ be a set of positive upper density. Suppose that $P_1,P_2,..., P_k\in \mathbb Z[X]$ are polynomials having zero constant terms. We show that the set $E\cap (E-P_1(p-1))\cap ... \cap (E-P_k(p-1))$ is non-empty for…

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set $I$, then it is true for all of $I$. This paper describes some old and new results on infinite sets of linear and…

It is shown that the set of palindromes is an additive basis for the natural numbers in any base. Specifically, we prove that every natural number can be expressed as the sum of $O(d)$ palindromes in base $d$.