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Math Help - addition of power numbers

  1. #1
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    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).
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  2. #2
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    Quote Originally Posted by linshi View Post
    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.
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  3. #3
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    I'm surprised that (2) is a famous problem...

    So is (1) also an unsolved problem too?
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