engleski

On the largest element in D(n)-quadruples

Let n be a nonzero integer. A set of nonzero integers {; ; ; ; ; ; a1, . . . , am}; ; ; ; ; ; such that aiaj + n is a perfect square for all 1 ≤ i < j ≤ m is called a D(n)-m-tuple. In this paper, we consider the question, for a given integer n which is not a perfect square, how large and how small can be the largest element in a D(n)- quadruple. We construct families of D(n)- quadruples in which the largest element is of order of magnitude |n|^3, resp. |n|^(2/5).

D(n)-quadruples, Diophantine equations, elliptic curves

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