Mersenne: Re: Munich prime party report

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Steve Harris (with Alex Kruppa for moral support and more importantly, to
run the tap and make sure the Steins stayed filled) wrote:

>We finally got the picture, text, and translations approved by all involved,
>so here is our report on the Munich chapter of the prime party of 7 december
>(at least as much as we can remember) :
>
>http://www.sheeplechasers.org/prime/muenchen/

Hey, thanks for the picture and explanatory text. Lends new meaning to
the phrase "mugging for the camera."

The caption accompanying the picture says:

>We discussed many things, both Mersenne and non-Mersenne. Alex tried to
>calculate ln(750000) with only pen, paper and beer

...not necessarily in that order of importance, I presume.

>and came up with an answer of "about 14" and was disappointed later
>to find out it was actually 13.5278... which Steve thought was not so bad.

Alex, I hope you weren't using a Taylor series for this - that would need
a pretty big napkin before it started converging!

>We discussed the observation (not a conjecture) that so far no two
>consecutive gaps between Mersenne primes (in terms of exponent ratio)
>were greater than two

Personally, I don't think there's anything special about a ratio of 2.
If we find no primes between M#38 and (what is currently) M#39, that
will represent two consecutive gaps which are very nearly 2 (one slightly
larger, one slightly smaller). For simplicity, let's call a triplet of
exponents which correspond to 2 consecutive gaps > 2 a "2-2 gap."
Here's an exercise for our readers to try during halftime of whichever
New Year's day foot-bowl games they might be watching:

Given a sequence of positive integers with the statistical property
that consecutive terms have a ratio of x_{n+1}/x_{n} = 1.45 on average
(This is close to the ratio predicted for Mersenne prime exponents
from the theoretical argument based on the known form of factors)
but otherwise random - to be more specific about the random nature,
assume a Poisson probability distribution

P(x) = exp(-lambda) * lambda^x / x! , with lambda = 1.45 .

How many terms (on average) are needed
before the probability of a 2-2 gap is >= 50%? Hw many such that the
probablity is >= 90%? How do those 2 numbers compare with the number
of know Mersenne primes?


>Alex said there was a German tradition, "auf ex" where, when toasting a
>special occasion, everyone's glasses must be emptied. Steve's not so sure
>about that one; what's referred to as a "glass of beer" in Germany is more
>commonly known as a "pitcher" in the U.S.

"auf ex" is German for (use your best Dick Vitale, sportscaster
extraordinaire, accent here) "bottoms up, baybee!" In one's quest
for knowldege, one should leave no Stein unturned. :)

>In case there is no new mersenne prime found in the near future, we plan to
>make this an annual event - or semiannual, or quarterly, or however often we
>can get together.

Speaking for the California chapter, a get-together to celebrate each new
prime (assuming we don't run into a huge gap or find two within weeks of
each other) seems like a sensible approach.

>We did decide that making it a daily event was totally out
>of the question :-)

Have Alex explain the meaning of the word "Zecher" to you. Like the
proverbial Eskimos and their 50 words (or however many) for snow,
I find it says a lot about a culture that they would even have a special
word for such a person. (For our non-German-speaking readers, the closest
english word I can muster that means roughly the same thing is "sot,"
but Zecher has less of a drunken fool connotation, and more of a
person-whose-professional-trade-it-is-to-drink-mass-quantities-of-
alcoholic-beverages.)

Happy new year, all!

-Ernst