(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.
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).