  In this semi-expository article we solve the diophantine equation $m^5+(4
\cdot 5^4 b^4)mn^4 - n^5=1$ for all integers $b \neq 0$. This gives an example
of a family of quintic Thue equations that can be solved completely by using
nothing more than Skolem's $p$-adic method. We also give a general introduction
to Skolem's method from a modern perspective.
