our first "dry" year in four years
 

From 2003 to 2006, GIMPS discovered at least one Mersenne prime each year. However, we didn't discover one in 2007. On the bright side, the longer we wait for somrthing, the better that "something" will be. When we discover the 45th Mersenne prime, it will likely be [i]very[/i] large.

That doesn't mean that we won't discover it tomorrow, though. :D

[QUOTE=ixfd64;121899]From 2003 to 2006, GIMPS discovered at least one Mersenne prime each year. However, we didn't discover one in 2007. On the bright side, the longer we wait for somrthing, the better that "something" will be. When we discover the 45th Mersenne prime, it will likely be [i]very[/i] large.

That doesn't mean that we won't discover it tomorrow, though. :D[/QUOTE]
The question is whether the first 10M-digit prime will be a Mersenne. I predict it will be of the form k*2^n+c, and be announced very soon.

[quote=jasong;122028]The question is whether the first 10M-digit prime will be a Mersenne. I predict it will be of the form k*2^n+c, and be announced very soon.[/quote]
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?

Several trends point to the middle 1/3 of '08 as being a likely point in time.
If they run ~18 months apart and the last was at the end of '06...

[quote=Mini-Geek;122037]The largest known prime has been a Mersenne prime since digital computers began[/quote]
With notable exceptions in 1951 and 1989
[URL]http://primes.utm.edu/notes/by_year.html#2[/URL]

[quote=Uncwilly;122038]Several trends point to the middle 1/3 of '08 as being a likely point in time.
If they run ~18 months apart and the last was at the end of '06...[/quote]
This "reasoning" is wrong.
If the expected time per prime were 18 months (and currently it is
nearer 4 years), it is the expected wait from "now", no matter how
much time has elapsed since the last prime.

Think of how many throws of a die till a six turns up.

I've been wondering how likely we are to see a maximal gap between M44 and M45. Obviously the whole discussion has to be subject to eventual double checks, but the leading edge for first time checks now has exponents over 7½ million past M44. So naively one* would now expect M45 to be more than 40,111,751.

Richard

* this 'one' anyway

[quote=Richard Cameron;122061]I've been wondering how likely we are to see a maximal gap between M44 and M45.[/quote]

I'm curious about what you mean with "maximal gap". As far as our knowledge to date is concerned, the gap between two consecutive mersenne primes could be arbitrarily large. And indeed, no-one as far as I know has managed to prove that there are infinitely many Mersenne primes or even that there are any more beyond what we have already found.

If GIMPS doesn't find any more mersenne primes in the forseeable future, the project is nonetheless continually extending the boundary of how far we know which Mersenne numbers are prime and which are composite. With that we are very gradually increasing our knowledge of their distribution, though this process is painfully slow. Some new breakthrough in the theory of how to test these numbers is probably the best hope for any significant progress now, I think. That said, discovering M45 in the coming months or years would be a very nice boost for morale.

[quote=Richard Cameron;122061]I've been wondering how likely we are to see a maximal gap between M44 and M45. Obviously the whole discussion has to be subject to eventual double checks, but the leading edge for first time checks now has exponents over 7½ million past M44. So naively one* would now expect M45 to be more than 40,111,751.

Richard

* this 'one' anyway[/quote]

Very likely. But what the heuristics say is that we expect
the next exponent to be ~1.5 times the last one, so the
expected "arithmetical" gap increases by 1.5 each time.
Note that the biggest ratio so far between exponents is that
from M127 (Lucas 1876) to M521 (Robinson with help from Lehmer
1952). Let's hope we aren't due to beat this record:smile:

David

[QUOTE=Brian-E;122064]I'm curious about what you mean with "maximal gap". [/QUOTE]

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.

Davieddy: you are right, the ratio Mn/Mn-1 is probably a more meaningful comparator.
[QUOTE=Davieddy]let's hope we aren't due to beat this record [/QUOTE]

nor the record 897 days between M38 and M39

Richard

We should hope to get two records in one shot :evil:

This might sound good on a press release: "After a record 1027 days since the last discovery, GIMPS has found a new record prime ..."