Bounds for Diophantine quintuples

A set of m positive integers {a1,...,am} is called a Diophantine m-tuple if the product of any two elements in the set increased by one is a perfect square. The conjecture according to which there does not exist a Diophantine quintuple is still open. In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a 3a; moreover, b > max{21 a, 2 a3/2} in case c>a+b+2(ab+1)1/2