  We consider Diophantine quintuples $\{a, b, c, d, e\}$. 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
$1.9\cdot 10^{29}$ Diophantine quintuples.
