On the existence of S-Diophantine quadruples

Abstract

Let S be a set of primes. We call an m-tuple (a 1 , . . . , am) of distinct, positive integers S-Diophantine, if for all i = j the integers s i,j := a i a j + 1 have only prime divisors coming from the set S, i.e. if all s i,j are S-units. In this paper, we show that no S-Diophantine quadruple (i.e. m = 4) exists if S = {3, q}. Furthermore we show that for all pairs of primes (p, q) with p < q and p ≡ 3 mod 4 no {p, q}-Diophantine quadruples exist, provided that (p, q) is not a Wieferich prime pair.2010 Mathematics Subject Classification. 11D61, 11D45.

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

“…Szalay and Ziegler [11] conjectured that if |S| = 2, then no S-Diophantine quadruple exists. This conjecture has been confirmed for several special cases by Luca, Szalay and Ziegler [11,12,13,16,5]. 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 S-Diophantine quadruples, if there exist any.…”

Diophantine triples with distinct binary recurrences

“…This was confirmed to be so in [7] in a stronger form and in [3] in a quantitative form. See [9], [17], [21], [22], [23] for more results in this direction. A different popular variation is when the squares are replaced by terms of a given binary recurrence.…”