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Diophantine quintuples containing two pairs of conjugates in some quadratic fields

In this paper, we describe constructions of Diophantine quintuples of the special form in rings Z[√D] for certain positive integer D. The term ”special form” refers to Diophantine quintuples of the form {; ; ; ; e, a+b√D, a-b√D, c+d√D, c- d√D}; ; ; ; , where a, b, c, d, e are integers. Also, we assume these quintuples contain two regular Diophantine quadruples.

Diophantine quintuples, regular Diophantine quadruples, quadratic fields

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