Doubly regular Diophantine quadruples
Abstract
For a nonzero integer n, a set of m distinct nonzero integers a1,a2,...,am such that ai aj + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n1)-quadruples and D(n2)-quadruples with distinct non-zero squares n1 and n2.
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