Odd perfect numbers

Importance: High ✭✭✭
Author(s): Ancient/folklore
Subject: Number Theory
Keywords: perfect number
Recomm. for undergrads: yes
Posted by: azi
on: September 27th, 2008
Conjecture   There is no odd perfect number.

There is substantial literature on the problem. Most proceeds from a study of the multiplicative function $ \sigma_{-1}(n)=\sigma(n)/n $ where the conjecture can be stated: $ \sigma_{-1}(n)=2 $ implies that $ n $ is even.


* indicates original appearance(s) of problem.

limiting divisors

My idea is to assume that the OPN is divisible by a prime number (e.x. 3) then use the properties of perfect numbers to figure out other numbers the OPN is divisible by.

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