The generalized Fermat equation has been solved for many signatures. But, I can't find a determination that the signature $[p,q,r]=[4,5,7]$ has no solutions. Is this signature still an open problem?
$\chi=1/4+1/5+1/7<1$
$x^p+y^q=z^r$
$gcd(x,y,z)=1$
$p,q,r\ge3$
The equation has only finitely many solutions with a bound given by the Darmon-Granville Theorem. Many other signatures have already been eliminated like $$ [p,p,p], [3,5,5], [4,5,5], [3,4,7], $$ and others.
Laisham and Shorey's 2011 result in their theorem 4 is that assuming Baker's conjecture there are no solutions to anything not in $$ \{[3,5,] : 7≤ ≤ 23, \text{ prime}\}∪\{[3,4,]: \text{ prime}\} . $$ This immediately implies at least a conditional elimination for $[4,5,7]$.