Difference between revisions of "Twin prime"

Revision as of 21:44, 21 April 2008

Twin primes are primes of the form $p$ and $p+2$ .

Main article: Twin Prime Conjecture

Use {{hatnote|text}} </noinclude>

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.

@@ Line 1: / Line 1: @@
-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>.
-, 5<br>
-, 7<br>
-, 13<br>
-, 19<br>
-, 31<br>
-, 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]].
+== 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.
-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]].
-{{wikify}}
 {{stub}}
 [[Category:Definition]]
 [[Category:Number Theory]]