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Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to be composite the difference must be greater than $p^2$ which is large compared to the average gap of size $p$.) This seems out of reach at the moment.

A somewhat weaker version asks if the difference is composite only finitely often. This, too, seems unassailable at present.

A much weaker version asks if $N(p\#+2)-p\#$ is prime infinitely often. Can this be proved with current technology?

Edit: to clarify, my question is about the third problem I mention: can it be solved, has it been solved, and if so what is the solution? It's 'obvious' that the second and third conjectures are correct (and the first one seems highly likely) but I'm interested in what's known.

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Can't help, but there is some data and some references at that might help someone who wants to take this one on. – Gerry Myerson Jul 6 '11 at 5:38

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