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

TL;DR

This paper develops a practical algorithm to find all $S$-Diophantine quadruples when the set of primes $S$ has size three, advancing the computational methods for these special number tuples.

Contribution

It introduces a new algorithm specifically designed to identify all $S$-Diophantine quadruples for fixed prime sets of size three.

Findings

Successfully implemented the algorithm for $|S|=3$

Identified all $S$-Diophantine quadruples for specific prime sets

Enhanced computational techniques for Diophantine problems

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 $1\leq i < j\leq m$ the quantity $a_ia_j+1$ 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 $|S|=3$.