Related papers: Every integer can be written as a square plus a sq…

A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally,…

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is…

In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.

One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.

In this paper we prove that if $A$ and $B$ are infinite subsets of positive integers such that every positive integer $n$ can be written as $n=ab$, $a\in A$, $b\in B$, then $\displaystyle \lim_{x\to \infty}\frac{A(x)B(x)}{x}=\infty $. We…

Hardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $\mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by…

We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number of divisors of $n$ and $d_4(n)$ is the number of ways of writing $n$ as a product of four numbers. In doing so we make a slight improvement on the upper…

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime.…

The goal of this paper is to count the number of distinct functions of n variables, up to permutation of the variables, that can be constructed using each variable exactly once, without constants, using only the operations of addition,…

We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of…

We consider words $w$ over the alphabet $\Sigma=\{0,1,2\}$. It is shown that there are irreducibly square-free words of all lengths $n$ except 4,5,7 and 12. Such a word is square-free (i.e., it has no repetitions $uu$ as factors), but by…

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free…

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…

An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas…

Let $b \ge 2$ be an integer. Among other results, we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of $b$ cannot have simultaneously only few distinct prime factors and only few nonzero…

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d \in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such…

Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is square-free.