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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.
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