The number of Diophantine quintuples

A set {a1, ... ,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a,b,c} with a < b < c, the number of Diophantine quintuples {a,b,c,d,e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10276, which improves the bound 101930 due to Dujella