Diophantine quintuples containing triples of the first kind

摘要

We consider Diophantine quintuples {a, b, c, d, e}, sets of integers with a b c d e the product of any two elements of which is one less than a perfect square. Triples of the first kind are sets {A, B,C} with C ≥ B5. We show that there are no Diophantine quintuples {a, b, c, d, e} such that {a, b, d} is a triple of the first kind.