[QUOTE=davieddy;108225]According to the GIMPS status page, the expected number of primes
before exponent 79,300,000 is 2.08.
That gives a probability of 12.5% that no prime will be found before then.
Your requested confidence level is too high to be useful.

There is a 72% probability that M45 will be higher than 40,250,000,
and a 41% probability it will be higher than 50,000,000.[/QUOTE]

Any such "probabilities" are meaningless. M45 is eiither > 40,250,000 or
it isn't. etc.

[quote=R.D. Silverman;108227]Any such "probabilities" are meaningless. M45 is eiither > 40,250,000 or
it isn't. etc.[/quote]
While it's true that M45 is either (not ei[U][I][B]i[/B][/I][/U]ther) < or > 40,250,000, [I]we do not yet know[/I] which of the two it is, so we can still say "There is a 72% probability that M45 will be higher than 40,250,000, and a 41% probability it will be higher than 50,000,000." without it being completely meaningless.

EDIT: Even once we know M45, it still would not be completely meaningless to say "Knowing what was known on June 13, 2007, there was a 72% probability that M45 would be higher than 40,250,000, and a 41% probability it would be higher than 50,000,000."

[QUOTE=Mini-Geek;108230]While it's true that M45 is either (not ei[U][I][B]i[/B][/I][/U]ther) < or > 40,250,000, [I]we do not yet know[/I] which of the two it is, so we can still say "There is a 72% probability that M45 will be higher than 40,250,000, and a 41% probability it will be higher than 50,000,000." without it being completely meaningless.

EDIT: Even once we know M45, it still would not be completely meaningless to say "Knowing what was known on June 13, 2007, there was a 72% probability that M45 would be higher than 40,250,000, and a 41% probability it would be higher than 50,000,000."[/QUOTE]

No. You can NOT make such a statement.

(1) We don't even know if M45 exists.
(2) Even if there are infinitely many, we do not know their *distribution*.
(3) Even if the *asymptotic* distribution were known, M45 is a long way
from infinity. --> there is no way to know how accurate such a probability
estimate might be.

I think it's useful to frame the question in terms, of "if you were a Vegas oddsmaker, how would set odds on M45?" In that sense:

[QUOTE=R.D. Silverman;108235](1) We don't even know if M45 exists.[/QUOTE]
If you were an oddsmaker, based on the known data, you wouldn't blink at the lack of proof in this regard. And one can easily formulate probabilistic contests in a way that does not require M45 to exist.

[QUOTE](2) Even if there are infinitely many, we do not know their *distribution*.[/QUOTE]
We have some pretty well-founded heuristics that appear to describe the known data quite well.

[QUOTE](3) Even if the *asymptotic* distribution were known, M45 is a long way
from infinity. --> there is no way to know how accurate such a probability
estimate might be.[/QUOTE]
Again, try to think of things the way a bookmaker would. Of course this problem is a bit different than (say) a horse race, in that we have to wait to see if a new prime is discovered to validate the betting. So an oddsmaker would likely frame bets as "odds of a new prime with p < 50M," because that doesn't require M45 to exist, and can be validated or refuted in finite time.

Often in this business of hunting large primes or factoring composite numbers we want to estimate our "probability" of success. To take an example from Bob's paper with Sam Wagstaff, when deciding when to switch over from ECM to NFS, we consider the "probability" that ECM will find a factor in a certain range so as to minimize the expected time needed for factoring. Of course, if the factor does not exist, the probability that ECM will be successful is miniscule, and other times, a little more ECM work above the optimal amount will find a factor more easily than by using NFS. Obviously, knowledge about the [B]true probability[/B] (0 or 1) of a factor existing in a certain size range would make factoring more efficient, but we don't have that knowledge and so we treat factor size as a random variable. The overall strategy is justified over a large number of factorizations, but in any one particular case, there is an element of gambling to it.

When Prime95 gives a "probability" that a LL-candidate is prime, of course this is not a probability in a strict mathematical sense, it is a Bayesian estimate based on the factored level of the candidate and certain assumptions on the distribution of Mersenne primes that seem to correlate well with the known cases. To say that because we don't know anything about the next Mersenne prime, that we are unjustified in making estimates at where it may be found ignores the premise that we expect that, over a large number of cases, that we will be successful in estimating the total number of new Mersenne primes found.

Maybe Mr. Silverman simply has a problem with how things are phrased. Although, in my mind, EVERY circumstance has a 0 or 1 probability of happening, it's simply a matter of possessing infinite knowledge. Even with quantum mechanics, the lack of quantifiability may be a temporary illusion.

[quote=davieddy;108225]According to the GIMPS status page, the expected number of primes
before exponent 79,300,000 is 2.08.
That gives a probability of 12.5% that no prime will be found before then.
Your requested confidence level is too high to be useful.
[/quote]

How valid is my naive application of the Poisson distribution here?
i.e. P(0)=e[sup]-2.08[/sup]

I took a little stab at predicting where M45 may occur. NB: This is based on some fairly rough calculations.

For M1-M44 based on Log2 (nth Mersenne Prime) the mean change is .5588
For M1-M44 the largest occurrences of variability to the mean are -1.1644 (M16) to 1.0264 (M26)

On the assumption that the next MP found (if it exists) will also fit within the historical variability it could be anywhere between:
2^24.423 (22,493,424)
2^26.613 (102,638,731)

Obviously this is pretty wide.

The distribution of variability is completely inconsistent with no resemblance of a bell curve. So I just tried simply knocking off the low 10 & high 10 extremes leaving the 23 least variable from the mean giving these results:
2^25.088 (35,664,860)
2^26.039 (68,947,743)

If the MPs followed the mean line it would look like:
M44 - 2^25.028 (35,467,639)
M45 - 2^25.587 (50,402,780)
M46 - 2^26.146 (74,255,780)

So VERY, VERY, VERY loosely (and excluding the fact that many candidates above p=22,493,424 have already been processed) I would say the chances of finding M45 are:
[LIST][*]Approx 25% chance before p=35,664,860[*]Approx 50% chance before p=50,402,780[*]Approx 75% chance before P=68,947,743[*]Approx 100% chance before p=102,638,731[/LIST]
I hope this is of some interest to someone :ermm:

[QUOTE=rgiltrap;108730]I hope this is of some interest to someone :ermm:[/QUOTE]
What's of interest to me is why one would want to encourage a chipmunk to smoke, which is what appears to be happening in your avatar.

Surely that chipmunk is underage. :razz:

What worries me is.... what is in the pipe?!

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