Finiteness results for Diophantine triples with repdigit values

TL;DR

This paper proves that the set of Diophantine triples with values in the set of repdigits in a given base is finite, providing effective bounds for such triples.

Contribution

The paper establishes effective finiteness results for Diophantine triples with repdigit values in any base, advancing understanding of their structure and limitations.

Findings

Finiteness of Diophantine triples with repdigit values proven

Effective bounds for these triples derived

Results hold for any base g ≥ 2

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 $\mathcal D_g$.