Finding all $S$-Diophantine quadruples for a fixed set of primes $S$

Abstract

Given a finite set of primes $S$ and a $m$-tuple $(a_1,\dots ,a_m)$ of positive, distinct integers we call the $m$-tuple $S$-Diophantine, if for each $1\le i<j\le m$ the quantity $a_i{a}_j+1$ has prime divisors coming only from the set $S$. For a given set $S$ we give a practical algorithm to find all $S$-Diophantine quadruples, provided that $|S|=3$.