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

TL;DR

This paper demonstrates how Skolem's $p$-adic method can be effectively used to solve a family of quintic Thue equations, providing both specific solutions and a modern overview of the method.

Contribution

It presents a complete solution to a family of quintic Thue equations using Skolem's $p$-adic method and offers a modern perspective on this classical technique.

Findings

Successfully solves the specific family of equations for all integer parameters.

Illustrates the effectiveness of Skolem's $p$-adic method in solving higher-degree Thue equations.

Provides a modern introduction to Skolem's method for broader understanding.

Abstract

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.