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Albanian IMO TST 2010 Question 4

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May 22, 2010
algebrapolynomialfunctioninequalitiesnumber theoryprime numbersnumber theory unsolved

Problem Statement

With σ(n)\sigma (n) we denote the sum of natural divisors of the natural number nn. Prove that, if nn is the product of different prime numbers of the form 2k−12^k-1 for k∈Nk \in \mathbb{N}(Mersenne′sMersenne's prime numbers) , than σ(n)=2m\sigma (n)=2^m, for some m∈Nm \in \mathbb{N}. Is the inverse statement true?
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