A family of quintic Thue equations via Skolem's $p$-adic method

In this semi-expository article we solve the diophantine equation $m^5+(4\cdot 5^4{b}^4)m{n}^4-n^5=1$ for all integers $b\mathrm{\ne }0$. 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.