Difference between revisions of "Twin prime"

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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>
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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 [[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]].
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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.
  
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>.
 
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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[[Category:Definition]]
 
[[Category:Definition]]
[[Category:Number Theory]]
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[[Category:Number theory]]

Latest revision as of 13:27, 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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