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IMC 2014, Problem 4

Source: IMC 2014

July 27, 2016
IMCcollege contestsnumber theoryprime factorization

Problem Statement

Let n>6n>6 be a perfect number, and let n=p1e1⋅⋅⋅pkekn=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k} be its prime factorisation with 1<p1<⋯<pk1<p_1<\dots <p_k. Prove that e1e_1 is an even number. A number nn is perfect if s(n)=2ns(n)=2n, where s(n)s(n) is the sum of the divisors of nn.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
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