Statistically speaking ...
 

... albeit from a non-statistician

From the GIMPS Status page:
- expect about 2 more MP's (1.93) up to 79,300,000.
- to complete the range will require 22,004,934 P90 Years.
- in 2006 887,665 P90 years done
- in 2007 971,861 P90 years done
- assume in the future it follows Moore's Law (take the conservative: double every 2 years)
- 2008-2009: 22,004,934 - 1,943,722 = 20,061,212 remaining
- 2010-2011: 20,061,212 - 3,887,444 = 16,173,768
- 2012-2013: 16,173,768 - 7,774,888 = 8,398,880
- 2014-2015: Finished!!!! ... sometime in 2015.

So... "statistically speaking" 7 or 8 more years to finish up to 79,300,000 during which time we will find 2 more Mersenne Primes.

I foresee more "dry years" on the horizon ... but don't let that discourage you. :flex:

[QUOTE=Richard Cameron;122071]I meant largest (discovered) arithmetic gap in exponent. I agree that since we don't know the distribution of mersenne primes, we can't make much in the way of predictions beyond that.[/QUOTE]

Actually, the known data and plausible heuristics [mainly by Pomerance, Lenstra and [url=http://primes.utm.edu/notes/faq/NextMersenne.html]Wagstaff[/url]] indicate a pretty clear probability distribution, with the ratio of successive prime exponents averaging ~1.5, While the ratio for any two successive M-primes may be as close to 1 as one likes[sup]*[/sup] and [likely] as large as one likes, based on the known data and the aforementioned algebraic heuristics, it is also unlikely for the ratio between successive p to exceed [say] 4. Of course, given the current range and speed of the search, a factor of 4 would represent quite a long "dry spell", at least by the standards of Internet-DC-project time.

===

[sup]*[/sup][i]More precisely, as near unity as allowed by the requirement that if p prime and M(p) prime, then M(p+2) is the smallest next-candidate [and only if p+2 also prime, obviously][/i]

[QUOTE=ewmayer;122082][sup]*[/sup][i]More precisely, as near unity as allowed by the requirement that if p prime and M(p) prime, then M(p+2) is the smallest next-candidate [and only if p+2 also prime, obviously][/i][/QUOTE]And if the twin-prime conjecture holds ...

Paul

I think petrw1's estimate of Moore's Law holding for GIMPS is far too optimistic. Indeed, the data in his post goes against it.

[QUOTE]- in 2006 887,665 P90 years done
- in 2007 971,861 P90 years done[/QUOTE]

This is a growth rate of less than 10 percent. Assuming exponential growth, it would take over 7.6 years for speed to double.

My experience, if I remember correctly, from reading Primenet's main page is that speed has been growing much more slowly (if at all) since late 2005.

By the way, where did he get the data for the total amount of work done in 2006 and 2007?

[QUOTE=ewmayer;122082]Actually, the known data and plausible heuristics [mainly by Pomerance, Lenstra and [url=http://primes.utm.edu/notes/faq/NextMersenne.html]Wagstaff[/url]] indicate a pretty clear probability distribution, with the ratio of successive prime exponents averaging ~1.5...[/QUOTE]

thanks for this. i had a quick look on the prime pages, but couldn't find this bit on the distribution of mersenne primes. Serves me right for posting at work.

[QUOTE=jinydu;122113]I think petrw1's estimate of Moore's Law holding for GIMPS is far too optimistic. Indeed, the data in his post goes against it.

This is a growth rate of less than 10 percent. Assuming exponential growth, it would take over 7.6 years for speed to double.

My experience, if I remember correctly, from reading Primenet's main page is that speed has been growing much more slowly (if at all) since late 2005.

By the way, where did he get the data for the total amount of work done in 2006 and 2007?[/QUOTE]

I considered adding a second scenario based on the 2006 to 2007 increase --- yes, much below Moore's law --- but didn't, mostly because I thought it was a pretty small sample size (one year) to predict the future on.

As well there is a lot of buzz about the quads now so I am somewhat optomistic that in 2008 there will be a bigger increase.

As to my source ... I have been printing the status pages ( [url]http://www.mersenne.org/status.htm[/url] ) every month since October 2005 as close to month end as George's updates permit. So I can tell you that on Oct 25, 2005 there were a little over 24 Million P90 years remaining; 23,864,460 as of Dec 28, 2005; 22,976,795 as of Dec 31, 2006 and 22,004,934 as of Dec 31, 2007.

I also keep some status pages, here is my contribution:


[U]Date[/U]: [U]Yrs remaining[/U]:

03FEB2002 - 27,338,916 (the oldest I have)
01JAN2003 - 26,531,251
31DEC2003 - 25,701,037
10JAN2005 - 24,601,120
01JAN2006 - 23,848,134

It looks like we are essentially maintaining our rate over the years, but I think that is due to the fact that we are searching for factors at ever increasing bit levels, which means that much more work is needed to find them. This is particularly evident in LMH work, that accounts for the majority of the decrease in the "number of years remaining". So in fact we are investing a lot more power in the search. As the bit levels increase, though, I am not sure whether we will able to "keep the pace". As an example, going from 64 to 65 bits takes a long time, much more than the double of the time needed to go from 63 to 64 bits. Hence, when the LMHers start hitting that wall, the increase in computer power will probably not be sufficient to allow us to reduce the "years remaining" at the rate we are curently doing.

Does the increase in factoring difficulty really make that much of a difference? I thought that the bulk of GIMPS' computing power is being devoted to LL testing.

Also, does this mean that the statistics do not accurately take into account the increased bit levels required for larger exponents?

[QUOTE=Mini-Geek;122037]I doubt the first 10M-digit won't be Mersenne. The largest known prime has been a Mersenne prime since digital computers began, and currently the list of largest primes is solid Mersenne's up through number 6, and the next k*2^n+c is under 4M-digit. (No, correlation does not imply causation, but there is indeed a cause. i.e. the extremely efficient LL test.)
Technically, Mersenne's do fit into your k*2^n+c, with k=1 and c=-1, but I know that's not what you meant.
Is there a specific DC project you think will find it? Are they testing many 10M-digit's? How long does it take for each number?[/QUOTE]
I believe it's already been found, by someone in the prime community, someone who's proven themselves to be knowlegeable about this stuff. According to them the doublecheck has been running since the middle of December.

When I say soon, I mean before Valentine's Day, which is February 14th.

[QUOTE=jasong;122204]I believe it's already been found, by someone in the prime community, someone who's proven themselves to be knowlegeable about this stuff. According to them the doublecheck has been running since the middle of December.

When I say soon, I mean before Valentine's Day, which is February 14th.[/QUOTE]

You have obtained inside information? I can't find anything on a new prime record from 2007.

When GIMPS finds a residue of zero, primenet announces it
as "prime unverified" although they don't disclose the exponent.
What is your source Jason?