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February 10th, 2011, 05:31 PM  #21 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Particular isolated primes
It's an extremely difficult result. I believe it was first proved by Westzynthius in his 1931 paper "Über die verteilung der zahlen die zu den n ersten primzahlen teilerfremd sind". The result has been improved many times; the most recent I've seen is Maier & Pomerance 1990, "Unusually large gaps between consecutive primes". This just shows that there are infinitely many such, though, not that there are infinitely many with p prime. It seems quite likely that this is true but I don't know that anyone's bothered to publish a proof of it. 

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