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Prime Power Sum

Problem

For any given prime, p, prove that 2p + 3p 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, 2p + 3p, will always be odd.

All odd squares are congruent with 1 mod 4: (2k+1)2 = 4k2 + 4k + 1.

For p greater than or equal 2, 2p congruent 0 mod 4. So if the sum is to be a perfect square, 3p must be congruent with 1 mod 4.

However, 31 congruent 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 congruent 27 congruent 3 mod 4. Similarly, 35 congruent 3 mod 4. That is, each time we multiply by 32 it continues to be congruent with 3 mod 4. Therefore, 2p + 3p can never be a perfect square.

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

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