1. ## addition of power numbers

let a,b,c,d be positive integers.

(1) if a > b > 1, are there any solutions to a^b + b^a = c^d, other than (a,b)=(4,2) or (6,2)?

(2) if a,b > 1, are there any solutions to a^b + 1 = c^d, other than (a,b)=(2,3)?

I've checked using a PERL program, and found no other solutions for c^d < 10^15 (which is about the precision limit).

2. Originally Posted by linshi
let a,b,c,d be positive integers.

(1) if a > b > 1, are there any solutions to a^b + b^a = c^d, other than (a,b)=(4,2) or (6,2)?

(2) if a,b > 1, are there any solutions to a^b + 1 = c^d, other than (a,b)=(2,3)?

I've checked using a PERL program, and found no other solutions for c^d < 10^15 (which is about the precision limit).
Of course, you also want d > 1.

(2) is Catalan's conjecture, an open problem for many years. It was finally proved by the Romanian mathematician Preda Mihăilescu, as recently as 2002, that (2,3) is the only solution.

3. I'm surprised that (2) is a famous problem...

So is (1) also an unsolved problem too?