Difference between revisions of "Twin prime"

 
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Two primes that differ by exactly 2 are known as [[twin primes]].  The following are the smallest examples:<br>
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'''Twin primes''' are pairs of [[prime number]]s of the form <math>p</math> and <math>p+2</math>.  The first few pairs of twin primes are <math>(3, 5), (5, 7), (11, 13), (17, 19), (29, 31)</math>, and so on.  Just as with the primes themselves, twin primes become more and more sparse as one looks at larger and larger numbers.
3, 5<br>
 
5, 7<br>
 
11, 13<br>
 
17, 19<br>
 
29, 31<br>
 
41, 43<br>
 
  
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]].
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== Twin Prime Conjecture ==
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{{main|Twin Prime Conjecture}}
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The [[Twin Prime Conjecture]] asserts that there are infinitely many pairs of twin primes. It is not known whether this statement is true.
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{{stub}}
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[[Category:Definition]]
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[[Category:Number theory]]

Latest revision as of 14:27, 21 July 2009

Twin primes are pairs of prime numbers of the form $p$ and $p+2$. The first few pairs of twin primes are $(3, 5), (5, 7), (11, 13), (17, 19), (29, 31)$, and so on. Just as with the primes themselves, twin primes become more and more sparse as one looks at larger and larger numbers.

Twin Prime Conjecture

Main article: Twin Prime Conjecture

The Twin Prime Conjecture asserts that there are infinitely many pairs of twin primes. It is not known whether this statement is true.

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