Finiteness results for Diophantine triples with repdigit values

Abstract

Let $g\ge 2$ be an integer and ${\mathcal{R}}_g\subset \mathbb{N}$ be the set of repdigits in base $g$. Let ${\mathcal{D}}_g$ be the set of Diophantine triples with values in ${\mathcal{R}}_g$; that is, ${\mathcal{D}}_g$ is the set of all triples $(a,b,c)\in {\mathbb{N}}^3$ with $c<b<a$ such that $ab+1,ac+1$ and $ab+1$ lie in the set ${\mathcal{R}}_g$. In this paper, we prove effective finitness results for the set ${\mathcal{D}}_g$.