engleski

Diophantine m-tuples with elements in arithmetic progressions

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 (mod n). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {; ; ; ; ; ; a, b, c}; ; ; ; ; ; such that a = b = c = 0 (mod n).

Diophantine m-tuples ; arithmetic progressions

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