Irregular Diophantine m-tuples and elliptic curves of high rank

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and with rank equal 8. 1 Diophantine m-tuples A set of m nonzero rationals {a1, a2, . . . , am} is called a (rational) Diophantine m-tuple if aiaj + 1 is a perfect square for all 1 ≤ i < j ≤ m (see [4]). The first example of a Diophantine quadruple was the set { 1 16 , 33 16 , 17 4 , 105 16 } found by Diophantus (see [3]). Let {a, b, c} be a Diophantine triple and let d = a + b + c + 2abc± 2 √ (ab + 1)(ac + 1)(bc + 1) . Arkin, Hoggatt and Strauss [1] proved that ad + 1, bd + 1 and cd + 1 are perfect squares. Let {a, b, c, d} be a Diophantine quadruple such that abcd 6= 1 and let e = (a+b+c+d)(abcd + 1) + 2abc + 2abd + 2acd + 2bcd± 2 √ (ab+1)(ac+1)(ad+1)(bc+1)(bd+1)(cd+1) (abcd− 1)2 . In [4] we proved that ae + 1, be + 1, ce + 1 and de + 1 are perfect squares. 1991 Mathematics Subject Classification: 11G05.