arXiv subjects

Ákos Pintér

Publications and source records attributed to Ákos Pintér.

4 recordsLinked to original sources

arXiv2026

On a character-twisted analogue of Schäffer's equation

Let $f$ be a positive integer, and let $χ$ be a primitive quadratic character of conductor $f$. Let $k$ be a positive integer, and write $B_k(χ,X)$ for the $k$-th Bernoulli polynomial corresponding to $χ$. Suppose $B_k(χ,X)$ is irreducible and of degree at least $2$. Then for 100% of positive integers $m$ divisible by $f$, the Diophantine equation \[ χ(1) \cdot (x+1)^k+χ(2) \cdot (x+2)^k+\cdots+χ(m) \cdot (x+m)^k \, =\, y^n, \] has no solutions with $x$, $y$, $n$ integers, and $n \ge 2$.

math.NT↗

arXiv2023

On power values of pyramidal numbers, II

For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation $\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$, for $6 \leq m \leq 100$. In this paper, we consider the question of higher powers, and find all solutions to the equation $\mathrm{Pyr}_m(x) = y^n$ in positive integers $x$, $y$, and $n$, with $n \geq 3$, and $5 \leq m \leq 50$. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.

math.NT↗

arXiv2023

On equal values of products and power sums of consecutive elements in an arithmetic progression

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.

math.NT↗

arXiv2015

Another generalization of a theorem of Baker and Davenport

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set $\{1, 3\}$ cannot be extended to a Diophantine quintuple. As a consequence of our main result, it is shown that the Diophantine pair $\{1, b\}$ cannot be extended to a Diophantine quintuple if $b-1$ is a prime.

math.NT↗