mathschallenge.net

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 2² + 3² = 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: (2k+1)² = 4k² + 4k + 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, 3¹ 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 3² to get the next odd power, 3³ 27 3 mod 4. Similarly, 3⁵ 3 mod 4. That is, each time we multiply by 3² it continues to be congruent with 3 mod 4. Therefore, 2^p + 3^p can never be a perfect square.