Quintic diophantine equation $x^5+y^5=7z^5$

Question

Are there any non-zero integer solutions to the equation $x^5+y^5=7z^5$? I am unsure how to approach this.

Answer 1

This question has been solved on Mathoverflow.

Solution by Michael Stoll

The integer points are given by $(x,-x,0)$ for $x\in \mathbb Z$.

The trick is to reduce the equation to a form that has been studied more extensively. For any solution $(x,y,z)$ to the equation with $z\not=0$, let $X=-xy/z^2$ and $Y=\frac{x^5}{z^5}-\frac{7}{2}$. Then $Y^2-\frac{49}{4}=X^5$, which is a hyperelliptic curve, and there are techniques for finding all rational solutions to such an equation.