Difference between revisions of "Twin prime"
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| − | It is not known whether or not there are infinitely many pairs of twin primes. A natural attempt to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes <math>B=\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\cdots</math>. If <math>B=\infty</math>, then there would be infinitely many twin primes. However, it turns out that <math>B<\infty</math>, which proves nothing. The number ''B'' is called [[Brun's constant]]. | + | It is not known whether or not there are infinitely many pairs of twin primes. A natural attempt to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes: <math>B=\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\cdots</math>. If <math>B=\infty</math>, then there would be infinitely many twin primes. However, it turns out that <math>B<\infty</math>, which proves nothing. The number ''B'' is called [[Brun's constant]]. |
Revision as of 12:40, 26 June 2006
Two primes that differ by exactly 2 are known as twin primes. The following are the smallest examples:
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43
It is not known whether or not there are infinitely many pairs of twin primes. A natural attempt to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes: 
. If 
, then there would be infinitely many twin primes. However, it turns out that 
, which proves nothing. The number B is called Brun's constant.
