- --- In primenumbers@yahoogroups.com, Phil Carmody wrote:

> > http://www.mersenne.org/

Wow, indeed. Big congrats to George, Curtis and colleagues.

> Wow.

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.

Of course we all suppose that there is an infinitude of

Mersenne primes. But in the announcement of such a fine feat

it might have been better to distinguish supposal from proof:

Now the best principles, excepting divine, and mathematical,

are but hypotheses, within the circle of which we may indeed

conclude many things, with security from error. But yet the

greatest certainty, advanced from supposal, is still but

hypothetical. So that we may affirm, things are thus and

thus, according to the principles we have espoused. But we

strangely forget ourselves, when we plead a necessity of

their being so in nature, and an impossibility of their

being otherwise.

Joseph Glanvill (1636-1680)

David > I found a small mathematical nit to pick in the press release:

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.

> 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.

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>- Ok, now you have me thinking outside of “lurk mode” about the largest known primes and the following question:

For lack of better terminology at this point, I’ll define MLP(n)+1 as the first n primes multiplied together plus one and the question comes to mind as to how often that number is composite?

The question is inspired from the proof of infinite primes where if the number of primes were finite, then multiplying them all together and adding one would not be divisible by any of those primes and is then prime or composite of factors that are not in the finite set. Is there a searchable keyword for these numbers that “more likely prime” than other numbers near them and how often are they composite?

James

From: Chris Caldwell

> I found a small mathematical nit to pick in the press release:

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.

> 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.

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.

[Non-text portions of this message have been removed] - On Tue, Feb 5, 2013 at 10:42 PM, James J Youlton Jr

<youjaes@...>wrote:

> **

s/o else already defined this as http://oeis.org/A006862

> For lack of better terminology at this point, I�ll define MLP(n)+1 as the

> first n primes multiplied together plus one

>

see the references there for more (terminology & partial answers to all of

your questions).

Maximilian

[Non-text portions of this message have been removed] - --- On Wed, 2/6/13, Chris Caldwell <caldwell@...> wrote:
> > I found a small mathematical nit to pick in the press release:

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.

> > 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>

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.

Phil

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