Polignac's Conjecture

Importance: High ✭✭✭
Subject: Number Theory
Keywords: prime
prime gap
Recomm. for undergrads: no
Posted by: Hugh Barker
on: January 10th, 2011

\begin{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. \end{conjecture}

In particular, this implies:

\begin{conjecture} Twin Prime Conjecture: There are an infinite number of twin primes. \end{conjecture}


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*[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.


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


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

There's a slightly compressed version of this proof here:


Probably better to refer to this one as it is more focused.

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