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

Abstract

Given a finite set of primes S and an m-tuple $$(a_1,\ldots ,a_m)$$ ( a 1 , … , a m ) of positive, distinct integers we call the m-tuple S-Diophantine, if for each $$1\le i < j\le m$$ 1 ≤ i < j ≤ m the quantity $$a_ia_j+1$$ 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 $$|S|=3$$ | S | = 3 .