Diophantine [formula omitted]-tuples with elements in arithmetic progressions

Abstract

In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n≥3 there does not exist a Diophantine quintuple {a,b,c,d,e} such that a≡b≡c≡d≡e(modn). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {a,b,c} such that a≡b≡c≡0(modn).