Related papers: Square-full values of quadratic polynomials

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta <…

A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an…

Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More…

For every positive integer $n$, the quantum integer $[n]_q$ is the polynomial $[n]_q = 1 + q + q^2 + ... + q^{n-1}.$ A quadratic addition rule for quantum integers consists of sequences of polynomials $\mathcal{R}' =…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $f \in \mathbb{F}_q[x_1, \dots, x_n]$ a polynomial in $n$ variables and let us denote by $N(f)$ the number of roots of $f$ in $\mathbb{F}_q^n$. %Many authors, such as Wei Cao and Kung…

A result of P\'olya states that every sequence of quadrature formulas $Q_n(f)$ with $n$ nodes and positive numbers converges to the integral $I(f)$ of a continuous function $f$ provided $Q_n(f)=I(f)$ for a space of algebraic polynomials of…

We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order $1$. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign…

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a…

We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992).…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…

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…

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are…

A \emph{quasi-polynomial} is a function defined of the form $q(k) = c_d(k) k^d + c_{d-1}(k) k^{d-1} + ... + c_0(k)$, where $c_0, c_1, ..., c_d$ are periodic functions in $k \in \Z$. Prominent examples of quasi-polynomials appear in…

We show that if a polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}$, where $g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]$, then $f$ can be…

In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at…

An efficient, when compared to exhaustive enumeration, algorithm for computing the number of square-free words of length $n$ over the alphabet $\{a, b, c\}$ is presented.

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*} where $p$…