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.

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.