--- On Wed, 2/6/13, Chris Caldwell <caldwell@...
> > I found a small mathematical nit to pick in the press release:
> > http://www.mersenne.org/various/57885161.htm
> > > there certainly are larger Mersenne primes
> > The certainty of that proposition remains unproven to
> > the best of my knowledge.
> Doesn't it depend on the universe of discourse? You
> are absolutely correct about "mathematically certainty"
> (e.g., proof). But if this is "certainty"
> in the sense that if we flip a fair coin a few thousand
> times we will certainly eventually get heads, then I think
> the statement is fine. Unproven, not even necessarily
> true, but as certain as most things in our lives.
> Wouldn't it be grand if there were no more
> Mersennes? That, and the reason behind it,
> would be a marvelous discovery! But without
> any such argument, I see another Mersenne as an unproven
> certainty. <grin>
Absolutely agreed. Because we don't have the mathematical smarts to either prove the finiteness or infiniteness of the set of Mersenne primes, either would be a great step forward.
In some ways, I'm sure GIMPS would be equally happy with either proof too. If it's proven infinite, then they know that they can happily keep crunching with the same keenness that they demonstrate presently (which is plenty). But if it's proven that there are no more, then what could be more fulfilling than knowing that you *did the whole task to completion*? (There is a whole range of mathematically-interesting discoveries between these two extremes, of course.)
Until then, all we have is heuristics, and I'm quite happy to map an experimentally-supported heuristic onto the word "certainty". And the huge experiment is supporting the heuristics very very well.
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