Specialization of Galois groups and integral points on elliptic curves

Abstract:

Let $n\not =0, \pm 4$ be an integer. We show that the Galois group of $x^5-10nx^2-24n$ is $A_5$ precisely when $|n|$ appears in the purely periodic continued fraction expansion $[ |n|, |n|, |n|, \ldots ]$ of odd positive integer powers of $(1+\sqrt {5})/2$; otherwise the Galois group is $S_5$. This shows that entries A002827 and A135064 of the On-Line Encyclopedia of Integer Sequences agree except for $n=4$. The proof involves determining all integral points of certain curves of genus 1 and 2. For integral points of an elliptic curve we handle that in two ways: via a computer algebra system and by a method of Tate.