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

Given a finite set of primes S and an m-tuple ( a 1 , … , a m ) 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 .