Related papers: Disquisitiones Arithmeticae and online sequence A1…

In this paper, we characterize the odd positive integers $n$ satisfying the congruence $\sum_{j=1} ^ {n-1} j^{\frac{n-1}{2}}\equiv 0 \pmod n$. We show that the set of such positive integers has an asymptotic density which turns out to be…

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with…

We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with…

Let $F$ and $G$ be integer polynomials where $F$ has degree at least $2$. Define the sequence $(a_n)$ by $a_n=F(a_{n-1})$ for all $n\ge 1$ and $a_0=0.$ Let $\mathscr{B}_{F,\,G,\,k}$ be the set of all positive integers $n$ such that $k\mid…

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erd\H{o}s, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also…

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod…

Consider a Gaussian stationary sequence with unit variance $X=\{X_k;k\in {\mathbb{N}}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_n=n^{-1/2}\sum^{n-1}_{k=0}f(X_k)$, where $f$ designates a…

Erd\"os conjectured the existence of an infinite Sidon sequence of positive integers which is also an asymptotic basis of order 3. We make progress towards this conjecture in several directions. First we prove the conjecture for all cyclic…

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that…

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand…

In the Equal-Subset-Sum problem, we are given a set $S$ of $n$ integers and the problem is to decide if there exist two disjoint nonempty subsets $A,B \subseteq S$, whose elements sum up to the same value. The problem is NP-complete. The…

Let $r(k,A,n)$ denote the number of representations of $n$ as a sum of $k$ elements of a set $A \subseteq \mathbb{N}$. In 2002, Dombi conjectured that if $A$ is co-infinite, then the sequence $(r(k,A,n))_{n \geq 0}$ cannot be strictly…

In 1997 H.G.Diamond gave a condition on Beurling's generalized prime numbers in order that the corresponding generalized integers have a density. We give a new proof of this condition (Theorem 1) and a proof that it is not necessary…

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set…

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+\beta)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results…

We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with…

Consider universal data compression: the length $l(x^n)$ of sequence $x^n\in A^n$ with finite alphabet $A$ and length $n$ satisfies Kraft's inequality over $A^n$, and $-\frac{1}{n}\log \frac{P^n(x^n)}{Q^n(x^n)}$ almost surely converges to…

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7…

Normal dynamics in a quasi-one-dimensional channel of length L (\to\infty) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W…

If T is an ergodic automorphism of a Lebesgue probability space (X,A,m), the set of coboundries B = db =T(b)+b with symmetric difference + form a subgroup of the set of cocycles A. Using tools from descriptive set theory, Greg Hjorth showed…