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).