The [Fermat-Catalan conjecture](http://en.wikipedia.org/wiki/Fermat%E2%80%93Catalan_conjecture) states that there are only finitely many sex-tuples $(a, b, c, d, e, f)$ of positive integers such that


(1) $a^d + b^e = c^f$,


(2) $\gcd(a, b, c) =1$,


(3) $\frac{1}{d} + \frac{1}{e} + \frac{1}{f} \lt 1$.


Here, I have a question.



> 
> **Question** : What is *known* and *unknown* about sex-tuple $(A,B,C,D,E,F)$ of positive integers such that
> 
> 
> (4) $A^D + B^E = C^F$,
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> 
> (5) $\gcd(A, B, C) =1 $,
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> 
> (6) $\frac{1}{D} + \frac{1}{E} + \frac{1}{F} \color{red}{\ge} 1$,
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> (7) $D\ge 2,\ E\ge 2,\ F\ge 2$,
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> 
> (8) $(D,E,F)\not =(2,2,2)$,
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> (9) $(D,E,F)\not =(3,3,3)$.
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> 
> 


**Examples** : $${10}^2+3^5=7^3,\ \ 433^2+143^3=42^4.$$
I would like to know any relevant references as well.



Examples that the Fermat-Catalan conjecture does not cover
There are three cases when $1/D+1/E+1/F=1$, which give $(D,E,F)=(2,3,6)$, $(4,4,2)$, and $(3,3,3)$. In these three cases the equations define curves that are in fact elliptic curves of rank $0$, and the only solutions are the obvious ones where one of $A$, $B$, $C$ is $0$. 


The case $1/D+1/E+1/F>1$ is much more interesting. Here all the solutions have been parametrized (there are infinitely many). The last case that was done was $(2,3,5)$:


J. Edwards, A complete solution to $X^2+Y^3+Z^5=0$, J. Reine Angew. Math. 571 (2004), 213–236. 


