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 , . . . , am) of positive, distinct integers we call the m-tuple S-Diophantine, if for each 1 ≤ i < j ≤ 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.