twin prime conjecture TwinPrimesTheNumberOfConjuncture 2003-01-07 04:45:00 2006-10-04 15:13:11 Conjecture twin prime constant twin primes % 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 Two consecutive odd numbers which are both prime are called {\it twin primes}, e.g. 5 and 7, or 41 and 43, or 1,000,000,000,061 and 1,000,000,000,063. But is there an infinite number of twin primes ? In 1849 de Polignac made the more general conjecture that for every natural number $n$, there are infinitely many prime pairs which have a distance of $2n$. The case $n=1$ is the twin prime conjecture. In 1940, Erd\H{o}s showed that there is a constant $c<1$ and infinitely many primes $p$ such that $q-p<c \ln{p}$ where $q$ denotes the next prime after $p$. This result was improved in 1986 by Maier; he showed that a constant $c < 0.25$ can be used. The constant $c$ is called the \emph{twin prime constant.} In 1966, Chen Jingrun showed that there are infinitely many primes $p$ such that $p+2$ is either a prime or a semiprime.