Difference between revisions of "Twin Prime Conjecture"
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| − | The ''' | + | The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[infinite]]ly many pairs of [[twin prime]]s, i.e. pairs of primes that differ by <math>2</math>. |
| − | == | + | == Failed Proofs == |
=== Using an infinite series === | === Using an infinite series === | ||
| − | One | + | One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show that the sum |
| − | Unfortunately, it has been shown that this sum converges to a constant | + | <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> |
| − | + | of the [[reciprocal]]s of twin primes [[diverge]]s, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant <math>B</math>, 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. | |
| − | 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:Conjecture]] | [[Category:Conjecture]] | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
Revision as of 13:09, 22 April 2008
The Twin Prime Conjecture is a conjecture (i.e., not a theorem) that states that there are infinitely many pairs of twin primes, i.e. pairs of primes that differ by 
.
Failed Proofs
Using an infinite series
One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of Dirichlet's Theorem. If one can show that the sum
of the reciprocals of twin primes diverges, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant 
, 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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