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Polignac's Conjecture
Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
In particular, this implies:
Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.
Bibliography
*[P] A. de Polignac, Six propositions arithmologiques déduites de crible d'Ératosthène. Nouv. Ann. Math. 8 (1849), pp. 423--429.
* indicates original appearance(s) of problem.
Flaw
On January 13th, 2011 Hugh Barker says:
OK, someone has spotted the inevitable flaw in the logic and pointed it out, so not worth looking after all (though feel free if you want to play "spot the error"...
Compressed version
On January 11th, 2011 Anonymous says:
There's a slightly compressed version of this proof here:
http://barkerhugh.blogspot.com/2011/01/twin-prime-proof-compressed-version.html
Probably better to refer to this one as it is more focused.
Drupal
CSI of Charles University
Link
I removed this link and its description from the problem, since it is now known to be incorrect. For future reference here it is: http://barkerhugh.blogspot.com/2011/01/twin-primes-and-polignac-conjecture.html