Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
(a) \quad 2x^5+3y^5=6z^5
$$
$$
(b) \quad x^5+3y^5=7z^5
$$


Here is a short motivation. Equation $ax^d + by^d=cz^d$ is trivial for $d=1$, solved by Lagrange for $d=2$, investigated in depth by Selmer <https://doi.org/10.1007/BF02395746> for $d=3$. For $d=4$, at least the case $a=b=1$ is well-studied, see e.g. Section 6.6 of Cohen's famous book "Tools and Diophantine Equations, Vol. 1". However, I cannot find any references about the case $d=5$. The listed equations are the smallest ones with $d=5$ for which I cannot find neither small solution $(x,y,z)\neq (0,0,0)$ nor local obstructions modulo small $p$ (not sure for until which $p$ I should check), hence the question. If you can point me any references for the case $d\geq 5$ (or $d=4$ with general $a,b,c$), this would also be interesting.



Existence of rational points on generalized Fermat quintics
Both curves have no rational points.


Curves of this shape map to genus 2 curves of the form $y^2 = a x^5 + b$
(one can make $a = 1$ if one likes), by quotienting out by the group
of automorphisms generated by $(x:y:z) \mapsto (\zeta x:\zeta^{-1} y:z)$
where $\zeta$ is a primitive fifth root of unity (one can pick any two
coordinates for the scaling to act on). For curves of this type, Magma can
fairly easily compute the Mordell-Weil group (group of rational points
on the Jacobian variety). In both cases, for the first curve I tried
(using the quotient as above) the Mordell-Weil group has rank 1.
One can then use Chabauty's method (also implemented in Magma) to determine
the set of rational points on the genus 2 curve. In each case, there
are three points, but they do not lift to rational points on the given
curve.


(EDIT:) Using the quotients via the action on $x$ and $z$ leads to
curves whose Jacobians have finite Mordell-Weil groups, leading
to a simpler computation.


The three points one finds on the quotient curves correspond
to $x = 0$, $y = 0$ and $z = 0$. So if these are the only points
there, then one can easily check that there are no rational
points on the original curves.


