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

\begin{conjecture} There is no odd \Def{perfect number}. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

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.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* 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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