Abstract
Given a finite set of primes S and an m-tuple (a_1,ldots ,a_m) of positive, distinct integers we call the m-tuple S-Diophantine, if for each 1le i < jle 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.
Highlights
Given an irreducible polynomial f ∈ Z[X, Y ] and a set A of positive integers we consider the product Π= f (a, b). a,b∈ A a =bIt is an interesting question what is the largest prime divisor P(Π ) of Π or alternatively what is the number of prime divisors ω(Π ) of Π in terms of the size of |A|
Even a rather efficient algorithm has been described by Szalay and Ziegler [13] that finds for a given set S = { p, q} of two primes all SDiophantine quadruples, if there exist any
We present the method only in the case that |S| = 3 obvious modifications to the algorithm would provide an algorithm that would work for any set S of primes with |S| ≥ 3
Summary
Given an irreducible polynomial f ∈ Z[X , Y ] and a set A of positive integers we consider the product It is an interesting question what is the largest prime divisor P(Π ) of Π or alternatively what is the number of prime divisors ω(Π ) of Π in terms of the size of |A|. In case of f (x, y) = x + y this question was intensively studied by several authors starting with Erdos and Turán [4] who established that in this case ω(Π ) log |A|. It was shown by Erdos, Stewart and Tijdeman [3] that this is essentially best possible.
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