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Prove n is square-free

Source: Mexico National Olympiad 2015 Problem 6

November 25, 2015
number theory

Problem Statement

Let nn be a positive integer and let d1,d2,…,dkd_1, d_2, \dots, d_k be its positive divisors. Consider the number f(n)=(−1)d1d1+(−1)d2d2+⋯+(−1)dkdkf(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k Assume f(n)f(n) is a power of 2. Show if mm is an integer greater than 1, then m2m^2 does not divide nn.
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