Is it ever the case that 2^m+3^m is a perfect power, whenever m is a positive prime number?

All perfect even powers are also squares.

All squares = 0 or 1 mod 4.

For m > 1, 2^m + 3^m = 1 or -1 mod 4, according to whether m is even or odd.

So m is even, say m=2n.

Then 2^m + 3^m = 2^2n + 3^2n = 4^n + 9^n = (-1)^n + (-1)^n mod 5 = 2 or -2 mod 5.

But all squares can only = 0, 1, or -1 mod 5.

So 2^m + 3^m can never be a perfect square, and thus not a perfect even power.

Posted by xdog on 2007-06-28 21:00:43