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.

Bibliography

% 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.

Comment viewing options

Select your preferred way to display the comments and click "Save settings" to activate your changes.