[QUOTE=Brian-E;211752]So why would the recent suggested cluster of Mersenne primes make the last year's lack more likely?[/QUOTE]

Because, as axn suggested, the base rate might not be the same as that experienced over the last 10 years.

If a gambler watches a hundred die rolls and calculates their average to be 3.6, she needs to assume that he's witnessed an unlikely event* or that the die is unbalanced. Based on his knowledge of other things (how trustworthy is the casino, etc.), she decides which to accept. In this case, as the gambler, I'm considering the possibility that the event "there are 9 Mersenne primes in such and such interval" is unlikely, and that the base expectation is lower than suggested by that interval.

I'm hesitant to consider early GIMPS periods, because they had very small candidates (relatively...) and few participants, and I'm not sure how well those cancel out. A better analysis (CPU-years per year, average size per year, linearize the difficulty, etc.) could no doubt remove that... but I haven't done that.


* What would you think? How unlikely is such an average over 100 rolls of a fair die? [SPOILER]29% to be that high or higher... not particularly unusual. In this case the gambler may well be right in supposing the dice to be fair.[/SPOILER]

Thanks CRGreathouse and axn for the clear explanations.

Yes, I see it now. We are dealing with a situation where the theoretical expected rate of occurrence of primes amongst Mersenne numbers is unknown, and our estimate of it on the basis of the total number which have been discovered so far in the range tested should be affected by evidence that our observations to date deviate from the norm. The recent cluster is evidence of that.

(Just summarizing what I understand from what you've said - please let me know if I still seem to be suffering from foggy thinking.:smile:)

Maybe someone has already done this before(or possibly multiple people have), but I used my inferior(for the forum) math skills to calculate the average base-2 logarithmic difference between the last 8 Mersenne exponents. and came up with the number .17667286.

So I tentatively predict that the next Mersenne prime exponent will be close to 48,200,000.

Who else came up with another number close to this before me? I'm pretty sure there's at least 1 or 2 other people that have done this same thing.

Edit: If I may go off on a fairly wild tangent here, I have a friend who tried sieving numbers of the form k*2^n+1, where 2^n+1 was a Fermat number. Sometimes he would find factors of 7 digits or more which would wipe out a huge chunk of the remaining equations in one fell swoop(as much as 1/3rd of the numbers in a sieve file). Maybe there are properties to the factors of the Mersenne numbers that we haven't discovered which could shed some light on why the last 8 Mersenne primes are so close together. Possibly even the sort of thing which could help find gigantic primes in general, since sieving is a major part of prime-finding.

Precisely, Brian. And thanks for the vote of confidence above ("you're obviously not someone who is likely to fall for the gambler's fallacy"). :smile:

Gratuitous flame-bait theory: I notice that the hook to the right on the base-2 logarithmic graph of the Mersenne primes starts on a number found a year or two after the ultra-hot P4 phenomenon started. I would suggest God, who exists outside of time, realized the prime numbers would start coming more slowly after this and tweaked the mathematical constants to make the occurence of Mersenne primes continue at the same approximate rate.

Praise the Lord!

[quote=jasong;212448]Gratuitous flame-bait theory: I notice that the hook to the right on the base-2 logarithmic graph of the Mersenne primes starts on a number found a year or two after the ultra-hot P4 phenomenon started. I would suggest God, who exists outside of time, realized the prime numbers would start coming more slowly after this and tweaked the mathematical constants to make the occurence of Mersenne primes continue at the same approximate rate.

Praise the Lord![/quote]If He exists outside of time, why did He wait until something was about to happen before "realizing" it and then changed things at almost the last moment?

I believe He extended His noodly appendage at the moment of creation to make sure everything worked out correctly.

Ramen!

[QUOTE=xilman;212451]I believe He extended His noodly appendage at the moment of creation to make sure everything worked out correctly.[/QUOTE]

noodly appendage? You mean the Flying Spaghetti Monster ( ref [url]http://www.venganza.org/[/url] )?

[quote=lfm;212455]noodly appendage? You mean the Flying Spaghetti Monster ( ref [URL]http://www.venganza.org/[/URL] )?[/quote]You didn't know that I'm a Pastafarian?

Paul

[quote=jasong;212364]
So I tentatively predict that the next Mersenne prime exponent will be close to 48,200,000.
[/quote]

Yeah, yeah, that estimation is really no secret. I would predict it in the same intervall ...

I'll go for 78M, found on January 2013.

Its now (Sep 13th 2010) 519 days since last prime found (Apr 12th 2009) which makes this the 4th largest gap between GIMPS primes: [URL="http://www.mersenneforum.org/showpost.php?p=211761&postcount=68"]post #68[/URL]

But still over a year until its a new gap record (Sep 27th 2011, 898 days), so lets hope we get a new prime soon :)