D(n)-quintuples with square elements

For an integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j+n is a perfect square for all 0<i<j<m+1, is called a D(n)-m-tuple. In this paper, we show that there are infinitely many essentially different D(n)-quintuples with square elements. We obtained this result by constructing genus one curves on a certain double cover of A^2 branched along four curves.