  Dujella and Peth\H{o}, generalizing a result of Baker and Davenport, proved
that the set $\{1, 3\}$ cannot be extended to a Diophantine quintuple. As a
consequence of our main result, it is shown that the Diophantine pair $\{1,
b\}$ cannot be extended to a Diophantine quintuple if $b-1$ is a prime.
