Examples that the Fermat-Catalan conjecture does not cover

Question

The Fermat-Catalan 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) $A^D + B^E = C^F$,

(5) $\gcd(A, B, C) =1 $,

(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) $(D,E,F)\not =(2,2,2)$,

(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.

Answer 1

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.

Answer 2

There are the cases $(D,E,F) = (2,2,n)$, $(2,3,3)$, $(2,3,4)$ and $(2,3,5)$ (up to permutation). In each case, there are finitely many parameterized families in terms of binary forms that give you all the solutions. This generalizes the well-known formulas for $(2,2,2)$. This is discussed, for example, in Henri Cohen's "Number Theory" books (Springer GTM 239, 240).

(I forgot the cases with $1/D + 1/E + 1/F = 1$; see Samir's answer for that.)