Bounds on the number of Diophantine quintuples

We consider Diophantine quintuples {a,b,c,d,e}{a,b,c,d,e}. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most 2.3⋅10292.3⋅1029 Diophantine quintuples.