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) $\frac{1}{D} + \frac{1}{E} + \frac{1}{F} \color{red}{\ge} 1$,
>
>(7) $D\ge 2,\ E\ge 2,\ F\ge 2$,
>
>(8) $(D,E,F)\not =(2,2,2)$,
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>(9) $(D,E,F)\not =(3,3,3)$.

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