On the existence of $S$-Diophantine quadruples

TL;DR

This paper investigates the existence of $S$-Diophantine quadruples, proving non-existence results for specific prime sets, notably for $S=\

Contribution

It establishes the non-existence of $S$-Diophantine quadruples for certain prime sets, extending understanding of Diophantine equations involving prime divisors.

Findings

No $S$-Diophantine quadruples exist for $S=\

No $\

Certain prime pairs do not admit $S$-Diophantine quadruples unless they are Wieferich pairs.

Abstract

Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.