On the sum of fourth powers in arithmetic progression

Abstract

We prove that the equation $(x-y{)}^4+x^4+(x+y{)}^4=z^n$ has no integer solutions $x,y,z$ with $\gcd (x,y)=1$ for all integers $n>1$. We mainly use a modular approach with two Frey $\mathbb{Q}$-curves defined over the field $\mathbb{Q}(\sqrt{30})$.

Cited by in corpus (2)

• Differences between perfect powers : prime power gaps
• $\mathbb{Q}$-curves and the Lebesgue-Nagell equation