Related papers: On a Conjecture about the Number of Solutions to L…

The Erd\"os-Moser conjecture states that the Diophantine equation $S_k(m) = m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers $k$ and $m$ with $k \geq 2$. We show that stronger conjectures about consecutive…

Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let \xi(n) denote the smallest positive integer b such that for each system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n, this…

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

Given the matrix equation ${\bf A X} + {\bf X B} + f({\bf X }) {\bf C} ={\bf D}$ in the unknown $n\times m$ matrix ${\bf X }$, we analyze existence and uniqueness conditions, together with computational solution strategies for $f \,:…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop…

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for…

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for…

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$,…

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…

A collection $\mathcal S$ of equivalence classes of positive definite integral quadratic forms in $n$ variables is called an $n$-exceptional set if there exists a positive definite integral quadratic form which represents all equivalence…

For integers $n,k,s$, we give a formula for the number $T(n,k,s)$ of order $k$ subsets of the ring $\mathbb{Z}/n\mathbb{Z}$ whose sum of elements is $s$ modulo $n$. To do so, we describe explicitly a sequence of matrices $M(k)$, for…

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set.…

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the…

The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The…

We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots,…

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…

In this note, we look at the diophantine equation $$ \prod_{i=1}^ta_i!=\prod_{j=1}^sn_i!, \quad n_1\geq \cdots \geq n_s\geq 2 \quad \textnormal{and}\quad n_1>a_1\geq a_2\geq\cdots \geq a_t\geq2. $$ \noindent Let $s<t$. Under the (explicit)…

As it follows from the theory of almost periodic functions the set of integer solutions $q$ to the Kronecker system $|\omega_{j} q - \theta_{j}| < \varepsilon \pmod 1$, $j=1,\ldots,m$, where $1,\omega_{1},\ldots,\omega_{m}$ are linearly…