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| Title of object: |
Germain prime |
| Canonical Name: |
GermainPrime |
| Type: |
Definition |
| Created on: |
2004-09-03 04:46:59 |
| Modified on: |
2004-09-03 04:46:59 |
| Classification: |
msc:11-XX |
Revision comment (for changes between this and next version):
| Changes for correction #6010 ('reclassify'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
A prime number $p$ is called a \emph{Germain prime} or \emph{Sophie Germain prime} if $2p+1$ is also prime.
The first few Sophie Germain primes are:
$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233,...$
A estimate for the number of Germain primes less than $n$ is $\frac{2cn}{\ln^2{n}}$, where $c$ is the twin prime constant.
It is conjectured that there are infinitely many Germain primes, but like the twin prime conjecture, this has not been proven. |
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