On the sum of fourth powers in arithmetic progression

Abstract: We prove that the equation $${ (x - y)⁴ + x⁴ + (x + y)⁴ = zⁿ }$$ 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 ${ \mathbb{Q}( \sqrt{30} ) }$.