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. | + | It is not known whether or not there are [[infinite]]ly many pairs of twin primes. The statement that there are infinitely many pairs of twin primes is known as the [[Twin Primes Conjecture]]. |
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| + | One proof that there are infinitely many primes involves showing that the sum of the reciprocals of the primes [[diverge]]s. Thus, a natural strategy 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>. | ||
| + | Unfortunately, it has been shown that this sum converges to a constant ''B'', known as [[Brun's constant]]. This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge. | ||
Revision as of 15:38, 29 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. The statement that there are infinitely many pairs of twin primes is known as the Twin Primes Conjecture.
One proof that there are infinitely many primes involves showing that the sum of the reciprocals of the primes diverges. Thus, a natural strategy to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes: 
.
Unfortunately, it has been shown that this sum converges to a constant B, known as Brun's constant. This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge.
