Thread: 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).

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.

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

So is (1) also an unsolved problem too?