Question

Prove that there are infinitely many distinct pairs (a, b) of co-prime integers `a>1` and `b>1` such that `a^b+b^a` is divisible by `a+b`?

Answer 1

See below.

Explanation:

Making `a=2k+1` and `b=2k+3` we have that

`a^b+b^a equiv 0 mod (a+b)` and for `k in NN^+` we have that `a` and `b` are co-primes.

Making `k+1=n` we have

`(2n-1)^(2n+1)+(2n+1)^(2n-1) equiv 0 mod 4` as can be easily shown.

Also can be easily shown that
`(2n-1)^(2n+1)+(2n+1)^(2n-1) equiv 0 mod n` so

`(2n-1)^(2n+1)+(2n+1)^(2n-1) equiv 0 mod 4n` and thus is demonstrated that for `a=2k+1` and `b=2k+3`

`a^b+b^a equiv 0 mod (a+b)` with `a` and `b` co-primes.

The conclusion is

... that there are infinitely many distinct pairs `(a, b)` of co-prime integers `a>1` and `b>1` such that `a^b+b^a` is divisible by `a+b`.