## Prime Power Sum

#### Problem

For any given prime, `p`, prove that 2^{p} + 3^{p} can never be a perfect square.

#### Solution

With the exception of `p`=2, for which 2^{2} + 3^{2} = 13 and is not square anyway, the sum, 2^{p} + 3^{p}, will always be odd.

All odd squares are congruent with 1 mod 4: (2`k`+1)^{2} = 4`k`^{2} + 4`k` + 1.

For `p` 2, 2^{p} 0 mod 4. So if the sum is to be a perfect square, 3^{p} must be congruent with 1 mod 4.

However, 3^{1} 3 mod 4, and as `p` will be odd (the case for `p` = 2 has been excluded), we note that multiplying both sides of the congruence by 3^{2} to get the next odd power, 3^{3} 27 3 mod 4. Similarly, 3^{5} 3 mod 4. That is, each time we multiply by 3^{2} it continues to be congruent with 3 mod 4. Therefore, 2^{p} + 3^{p} can never be a perfect square.

Problem ID: 191 (28 Nov 2004) Difficulty: 3 Star