twin prime conjecture

Two consecutive odd numbers which are both prime are called 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, Erdős 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 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.

单词	TwinPrimeConjecture
释义	twin prime conjecture Two consecutive odd numbers which are both prime are called 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, Erdős 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 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.
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