Euler proved that: $$2u(u^2+3v^2)=(2a(a^2+3b^2))^3$$ had no non-trivial solutions.
Several other mathematicians have proven that: $$x^5+y^5=z^5$$ Using a similar argument, it is easy to see when $x$ and $y$ are both odd that the above equation is equivalent to $$2u(u^4+10(uv)^2+5v^4)=z^5$$ when we set $x=u+v$ and $y=u-v$.
I wonder if we were able to show in the latter case that there exist integers $a$ and $b$ such that:$$z=2a(a^4+10(ab)^2+5b^4)$$ How could one conclude by infinite descent that the above equation has no non trivial solutions? I know it's hypothetical but it's worth inquiring. Stay safe everyone!