The BIG gap?
 

Searching for Mersenne Primes is in one sense similar to waiting for an earthquake in Los Angeles. When it comes it will destroy all the fun for a long time. A BIG gap in the series of exponents of Mersenne Primes will also destroy the fun in a similar way I suppose.

It has been more than two years now since the latest discovery of a Mersenne Prime on 2009 April 12. (this is the second biggest timegap between discoveries since 1996 September 3)

If we contemplate the following:

[FONT=Times New Roman][SIZE=3]M12 - 127[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M13 - 521[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]4,10[/SIZE][/FONT]

[FONT=Times New Roman][SIZE=3]M14 - 607[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M15 - 1,279[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]2,11[/SIZE][/FONT]

[FONT=Times New Roman][SIZE=3]M20 - 4,423[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M21 - 9,689[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]2,19[/SIZE][/FONT]

[FONT=Times New Roman][SIZE=3]M31 - 216,091[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M32 - 756,839[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]3,50[/SIZE][/FONT]

[FONT=Times New Roman][SIZE=3]M35 - 1,398,269[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M36 - 2,976,221[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]2,13[/SIZE][/FONT]

[FONT=Times New Roman][SIZE=3]M37 - 3,021,377[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M38 - 6,972,593[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]2,31[/SIZE][/FONT]

We can see that there have been quite a lot of big gaps relatively speaking. The "wave" is now around 53,000,000. And the progress is around 3M a year. So what is the probability that the next Mersenne prime will be larger than M100,000,000. That is, that we won't find the next Mersenne prime in 16-17 years?

It would be even worse if the relative gap between exponents would be as large as between M12 and M13. In that case the next Mersenne Prime would be M176,863,549. And I will most probably be dead when they find it.

Even if we find the next Mersenne Prime tomorrow it would be interesting to know if there will be more big gaps in the series of exponents of the Mersenne primes, and if so how frequently?

Speaking of "gaps" and "bunches" as if they mean anything concrete in the context of [URL="http://en.wikipedia.org/wiki/Poisson_distribution"]randomly[/URL]-distributed data like primes is sure to draw the criticism of some of the serious mathematicians around here. Gaps and bunches happen in any randomly-distributed data, but it means nothing. Trying to predict how big a gap will be, and whether or not that makes it part of a bunch, is only possible in the most general terms.
[QUOTE=aketilander;260849]The "wave" is now around 53,000,000. And the progress is around 3M a year. So what is the probability that the next Mersenne prime will be larger than M100,000,000.[/QUOTE]

For simplicity's sake, let's say the question is precisely, "what is the chance of finding a Mersenne prime with 50,000,000 < p < 100,000,000?".
According to [url]http://primes.utm.edu/notes/faq/NextMersenne.html[/url], (I recommend you read that page, it has lots of details on this subject) "The expected number of Mersenne primes 2[SUP][I]p[/I][/SUP]-1 with [I]p[/I] between [I]x[/I] and [I]2x[/I] is about [I]e[SUP]gamma[/SUP][/I]." [I]e[/I] is Euler's number, the base of the natural logarithm, about 2.718, and [I]gamma[/I] is the Euler–Mascheroni constant, about 0.5772. This is not known to be exact, (it's not proven that there are infinite Mersenne primes [I]or[/I] composites, let alone how they're distributed) but is conjectured.
That comes to ~1.781 expected primes per doubling of p. The probability of finding at least one prime given that is 1-e^(-1.781), or 83.15%. So the probability that the next largest Mersenne prime after 50M is over 100M is about 16.85%.
[QUOTE=aketilander;260849]Even if we find the next Mersenne Prime tomorrow it would be interesting to know if there will be more big gaps in the series of exponents of the Mersenne primes, and if so how frequently?[/QUOTE]

My educated guess: almost exactly as large and frequently as you'd expect in any [URL="http://en.wikipedia.org/wiki/Poisson_distribution"]Poisson distribution[/URL].

[QUOTE=aketilander;260849]

<senseless numerology deleted>

Even if we find the next Mersenne Prime tomorrow it would be interesting to know if there will be more big gaps in the series of exponents of the Mersenne primes, and if so how frequently?[/QUOTE]

The current conjecture is that the number of primes between
2^x and 2^(2x) is Poisson distributed with mean exp(gamma) as
x --> oo.

While there is no proof (and no hope of a proof for the near future)
the conjecture is supported by some fairly strong heuristics.

The conjecture gives an exact answer to your question.

[QUOTE=Mini-Geek;260852]

much common sense deleted...........


My educated guess: almost exactly as large and frequently as you'd expect in any [URL="http://en.wikipedia.org/wiki/Poisson_distribution"]Poisson distribution[/URL].[/QUOTE]

Bingo!

[QUOTE=R.D. Silverman;260858]Bingo![/QUOTE]

Bob:
You need to go see a doctor tomorrow...something is clearly wrong with you, your attitude is much too positive!!!!

It is also going to get a bit easier to find M48; TF has just had a two order of magnitude upgrade.

[QUOTE=R.D. Silverman;260858]Bingo![/QUOTE]

A bit of clarification for those who have not had a course in
probability theory.

The Poisson distribution is a (necessarily discrete) counting function.
It is the counting function that counts the number of arrivals in a given
random interval defined by the exponential waiting time density function
(also known as the Erlang distribution). The actual gaps between M_p are
determined by the Erlang distribution. The [b]count[/b] of the number
of M_p in any interval of given length is governed by the Poisson distribution.

[QUOTE=R.D. Silverman;260911]the exponential waiting time density function (also known as the Erlang distribution).[/QUOTE]

An Erlang distribution is a sum of identical exponential distributions. The number in the sum is called the shape parameter or the stage parameter. So an exponential distribution is a particular case of the Erlang distribution, with 1 stage. The counting function for Erlang distributions with more stages is not Poisson.

[QUOTE=Christenson;260880]Bob:
You need to go see a doctor tomorrow...something is clearly wrong with you, your attitude is much too positive!!!!

[/QUOTE]

Indeed.
A serious case.
If he carries on like this, he'll probably scratch me off
his copious "ignore" list by mistake.

David

[QUOTE=Christenson;260880]
It is also going to get a bit easier to find M48; TF has just had a two order of magnitude upgrade.[/QUOTE]

Yes but no need to wet your pants.

An extra 2 bits, say 68-70 (vintage years for me)
will find a factor 2 times in 70, and if no such factor is found,
increase the chance of primality accordingly.

David

BTW do you mean x2 or x10 when you say "order of magnitude"?

Whom the Gods Love Die Young
 

[QUOTE=aketilander;260849]
If we contemplate the following:

[FONT=Times New Roman][SIZE=3]M12 - 127[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]M13 - 521[/SIZE][/FONT]
[FONT=Times New Roman][SIZE=3]4,10[/SIZE][/FONT]

It would be even worse if the relative gap between exponents would be as large as between M12 and M13. In that case the next Mersenne Prime would be M176,863,549. And I will most probably be dead when they find it.

[/QUOTE]

Indeed.
Alan Turing had 1024 bits of RAM at his disposal, so missed M13.

Mersenne(???), Fermat, Euler and Lucas may also have had
to wait for the news of a new MP from heaven.

[URL="http://www.youtube.com/watch?v=AcWjQPezoJ0&feature=related"]
That'll be the Day[/URL]

David

[url]http://oeis.org/A002515[/url] seems to knock out a lot of them ( if it's as simple as in the title then over 100000 under 43112609) is there a 4n+1 equivalent ?