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| − | Two [[prime number | primes]] that differ by exactly 2 are known as '''twin primes'''. The following are the smallest examples:<br>
| + | '''Twin primes''' are primes of the form <math>p</math> and <math>p+2</math>. |
| − | 3, 5<br>
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| − | 5, 7<br>
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| − | 11, 13<br>
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| − | 17, 19<br>
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| − | 29, 31<br>
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| − | 41, 43<br>
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| − | 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 Prime Conjecture]].
| + | == Twin Prime Conjecture == |
| | + | {{main|Twin Prime Conjecture}} |
| | + | The statement that there are infinitely many pairs of twin primes is known as the [[Twin Prime Conjecture]], which has not been proven yet. |
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| − | One proof that there are infinitely many primes involves showing that the sum of the [[reciprocal]]s 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>.
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| − | 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 [[finite]]ly many twin prime pairs or that they are spaced "too far apart" for that [[series]] to [[diverge]].
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| − | {{wikify}}
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| | [[Category:Definition]] | | [[Category:Definition]] |
| | [[Category:Number Theory]] | | [[Category:Number Theory]] |
Revision as of 21:44, 21 April 2008
Twin primes are primes of the form 
and 
.
Twin Prime Conjecture
- Main article: Twin Prime Conjecture
The statement that there are infinitely many pairs of twin primes is known as the Twin Prime Conjecture, which has not been proven yet.
This article is a stub. Help us out by expanding it.
and 
.
