Related papers: Some new results on consecutive equidivisible inte…

A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based…

We show that there exist exactly 203 positive integers $N$ such that for some integer $d \geq 2$ this number is a $d$-digit palindrome base 10 as well as a $d$-digit palindrome for some base $b$ different from 10. To be more precise, such…

For a positive integer $n>1$ denote by $\omega(n)$ the maximal possible number $k$ of different functions $f_1,\dots,f_k:\mathbb{Z}/n\mathbb{Z}\mapsto \mathbb{Z}/n\mathbb{Z}$ such that each function $f_i-f_j,i<j$, is bijective. Recently A.…

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|,…

The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach,…

I introduce the problem of finding maximal sets of equiangular lines, in both its real and complex versions, attempting to write the treatment that I would have wanted when I first encountered the subject. Equiangular lines intersect in the…

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[\alpha p]$, $[\alpha p]+1$, where $p$ is prime and $\alpha>0$ is irrational algebraic number. We also establish an asymptotic…

Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$ where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive cubes on $E$…

For each positive integer $k$, we consider five well-studied posets defined on the set of Dyck paths of semilength $k$. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets.…

When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

Let $B$ be a finite set of natural numbers or complex numbers. Product set corresponding to $B$ is defined by $B.B:=\{ab:a,b\in B\}$. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence…

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…

We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

Given a set of $t$ words of length $n$ over a $k$-letter alphabet, it is proved that there exists a common subsequence among two of them of length at least $\frac{n}{k}+cn^{1-1/(t-k-2)}$, for some $c>0$ depending on $k$ and $t$. This is…

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint…

This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the…

A well-known result of Verstra\"ete \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of…