$5^m$, where m is any natural, can be expressed as the sum of two perfect squares? Prove that for all natural $m$, $5^m$ can be expressed as the sum of two perfect squares.
Also, prove that $5^m + 2$ can be expressed as the sum of three perfect squares.

You can prove the first statement by induction on $m$. $m = 1$: $5^1 = 5 = 1+4 = 1^2+2^2$. Assume $5^m = a^2+b^2$, we have: $5^{m+1} = (1^2+2^2)(a^2+b^2) = (a-2b)^2+(b+2a)^2$ is a quite well-known identity.
