On a character-twisted analogue of Schäffer's equation

Kálmán Györy; Vandita Patel; Ákos Pintér; Samir Siksek

On a character-twisted analogue of Schäffer's equation

Number Theory 2026-07-16 v1

Authors: Kálmán Györy , Vandita Patel , Ákos Pintér , Samir Siksek

Abstract

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

Keywords

schaffer's equation diophantine equations bernoulli polynomials quadratic characters irreducibility

Cite

@article{arxiv.2607.15090,
  title  = {On a character-twisted analogue of Schäffer's equation},
  author = {Kálmán Györy and Vandita Patel and Ákos Pintér and Samir Siksek},
  journal= {arXiv preprint arXiv:2607.15090},
  year   = {2026}
}