Difference between revisions of "Twin Prime Conjecture"
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The '''twin prime conjecture''' is a yet unproven conjecture that states that there are [[infinite]]ly many pairs of [[twin prime]]s. Twin primes are primes of the form <math>p</math> and <math>p+2</math>. | The '''twin prime conjecture''' is a yet unproven conjecture that states that there are [[infinite]]ly many pairs of [[twin prime]]s. Twin primes are primes of the form <math>p</math> and <math>p+2</math>. | ||
| + | |||
| + | == Possible Proofs == | ||
| + | === Using an infinite series === | ||
| + | One proof that there are infinitely many twin primes involves showing that the sum of the [[reciprocal]]s of twin primes [[diverge]]s. A strategy to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes: <center><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></center> | ||
| + | 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. | ||
{{stub}} | {{stub}} | ||
| + | [[Category:Theorem]] | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
Revision as of 20:47, 21 April 2008
The twin prime conjecture is a yet unproven conjecture that states that there are infinitely many pairs of twin primes. Twin primes are primes of the form 
and 
.
Possible Proofs
Using an infinite series
One proof that there are infinitely many twin primes involves showing that the sum of the reciprocals of twin primes diverges. A 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.
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