Mersenne Digest Tuesday, 16 March 1999 Volume 01 : Number 533
----------------------------------------------------------------------
From: "Brian J Beesley" <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 09:41:03 GMT
Subject: Re: Mersenne: LL testing
From: [EMAIL PROTECTED]
Date sent: Mon, 15 Mar 1999 23:32:30 EST
To: [EMAIL PROTECTED]
Subject: Mersenne: LL testing
> I'm curious, as to the nature of the proof that the LL test can definitively
> prove/disprove the primality of a Mersenne number. The resources I have are
> limited and don't go into depth. Before I search elsewhere, I was wondering if
> anyone on the list could help me:
>
> What were the original details of Lucas' test? And how did Lehmer modify it
> into its current form (at least I know how to perform the LL test in its
> current state).
>
> What are the details of the proof that Lucas' test definitively (dis)proves
> the primality of a Mersenne number? Does the proof change for the Lucas-Lehmer
> test?
Suggest you look at the following references, which I just found by
following links from Chris Caldwell's Web site.
J. W. Bruce, "A really trivial proof of the Lucas-Lehmer test," Amer.
Math. Monthly, April (1993) 370-371. (Proves sufficiency only. See
also M. I. Rosen "A proof of the Lucas-Lehmer test," Amer.
Math. Monthly, 95 (1988) 855-856.)
>
> In the LL test, we start with S(1) = 4. The Prime Page says we can use S(1) =
> 10 and certain other values depending on p. Can anyone clarify this?
Yes, there are a whole list of other starting values for which the LL
Test is valid. However, this don't really assist us a lot. The problem
is that, although the final residual will be zero if the tested number
is prime whichever of the starting values is used, if the number is
composite, the final residual will vary depending on the starting
value (but can't be zero). This makes cross-checking of results
difficult.
>
> To prove that M(127) (Or M127, whichever refers to 2^127 - 1, not the 127th
> Mersenne prime) is prime, did Lucas use his test by hand? I know he did it by
> hand, at the very least.
Very probably. Lucas was working in the time before efficient
calculating machines were in common usage (at any rate, in the
West; the Russians, Chinese and Japanese were using their
different varieties of abacus); Babbage's Difference Engine was
unique, specialised to its task and limited to 20 digits precision, so
would have been of little or no use to Lucas.
My interest in Mersenne primes dates back to a winter afternoon in
early 1965 when, incarcerated in the school library (sports were
abandoned due to the poor weather) I read about the LL Test
somewhere and decided to verify a few (very small) exponents. By
hand, I realised that neither a slide rule nor log tables would be any
help - and *no* schools even dreamt of posessing an electronic
computer!. Aged nearly 12, I was able to complete the verification
of M(31) i.e. 2^31-1 in about 80 minutes. I'm sure my arithmetic
has degenerated to the point where I couldn't possibly repeat the
exercise, but, it does show that hand-calculating the LL Test on
M(127) would be feasible (using "classical" techniques - long
multiplication etc - it would take approx. (127/31)^3 = 70 times as
long to LL test M(127) as M(31)). Tedious but definitely possible.
The fact that you, a $1000 computer and a suitably optimized
program could now do the test in a millisecond or two is irrelevant.
>
> Thanks. Please reply to the list, as if you have an Internet E-mail address,
> my software will auto-block you. AOL members need not worry.
> S.T.L.
Hmm. I prefer to send replies of specific interest to the enquirer
only. I guess there are some other people on the list who might be
interested enough to justify this broadcast?
Suggestion, would you consider changing your filter rules to let
through messages whose subject header contains the key word
"mersenne" irrespective of origin? I doubt many "spammers" would
deliberately change their message format to sneak by that rule.
Regards
Brian Beesley
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Roger Vives Miret <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 12:17:21 +0100
Subject: RE: Mersenne:Garbage
> >To be removed, simply call 1-800-600-0343 ext. 1746
And if their benefits consists on calling this phone number?
Or if I don't live in States?
Roger
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Chris Caldwell <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 10:13:10 -0600 (CST)
Subject: Re: Mersenne: LL testing
On Tue, 16 Mar 1999, Brian J Beesley wrote:
> Suggest you look at the following references, which I just found by
> following links from Chris Caldwell's Web site.
>
> J. W. Bruce, "A really trivial proof of the Lucas-Lehmer test," Amer.
> Math. Monthly, April (1993) 370-371. (Proves sufficiency only. See
> also M. I. Rosen "A proof of the Lucas-Lehmer test," Amer.
> Math. Monthly, 95 (1988) 855-856.)
> >
Perhaps the ideal reference, if you are serious, is the new text "E'douard
Lucas and Primality Testing" by Hugh C. Williams (Canadian Math. Soc.
Series of Monographs and Advanced Texts, v22, Wiley-Interscience, 1998).
This is an excellent history of primality proving--every math library
ought to buy a copy.
> > To prove that M(127) (Or M127, whichever refers to 2^127 - 1, not the 127th
> > Mersenne prime) is prime, did Lucas use his test by hand? I know he did it by
> > hand, at the very least.
>
> Very probably. Lucas was working in the time before efficient
> calculating machines were in common usage (at any rate, in the
> West; the Russians, Chinese and Japanese were using their
He used a 127 by 127 chessboard (see the above text, section 3.2: Lucas
and M127).
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: "Foghorn Leghorn" <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 14:38:22 EST
Subject: Re: Mersenne: RE: Meganet's Primality Code
> Without a proof Meganet's primality test is still a probabilistic
test,
>so why not use a free one. Here is one, no marketing hype, no
>endorsements and no "T-sequences"! Compile with GNU-MP.
>
>#include <stdio.h>
>#include "gmp.h"
>
>int main() {
>
> mpz_t N;
>
> mpz_init (N);
> printf ("Enter number:\t");
> mpz_inp_str (N, stdin, 10);
> if (mpz_probab_prime_p (N, 100)) printf ("Prime\n");
> else printf ("Composite\n");
>
>return (0);
>}
Better still, change the number of iterations in the probable-prime test
to 1000 and claim that your test is "1000% likely" to prove primality.
Wouldn't that be impressive on a corporate web page? :)
Get Your Private, Free Email at http://www.hotmail.com
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Jason Stratos Papadopoulos <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 15:42:23 -0500 (EST)
Subject: RE: Mersenne:Garbage
On Tue, 16 Mar 1999, Roger Vives Miret wrote:
> > >To be removed, simply call 1-800-600-0343 ext. 1746
>
> And if their benefits consists on calling this phone number?
> Or if I don't live in States?
The original spam came from a uunet IP address, and I complained to
[EMAIL PROTECTED] the day I got the ad.
jasonp
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: "David J. Zook" <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 14:43:05 -0600
Subject: Re: Mersenne:Garbage
Calling the 800 number is almost certainly not a good idea.
The fact that they have an 800 number setup indicates this is not
"accidental spam" from a newbie who is unaware of the rules for good
relations with potential customers on the Internet..
They will probably use your call to confirm that you are receiving the spam,
and may attempt to obtain further demographic information over the phone.
They can then market your email address to other spammers for a good price.
DejaNews has a service which provides an email address for use with USENET
groups. You send messages to the your favorite group with that special From:
address. DejaNews scans all incoming email to their special email domain for
spam, and deletes it. I have not actually used, so cannot say how effective
it is. They claim they make extensive efforts to spot and eliminate
spammers.
Jonathan A Zylstra wrote:
> -this was in the ad, so this might help
>
> >----------------------------------------------------------
> >To be removed, simply call 1-800-600-0343 ext. 1746
> >and leave your e-mail address. Thank you.
> >----------------------------------------------------------
> >>>>>CONTACT INFORMATION<<<<<
>
> Call us TOLL FREE today 1-877-SWIPE-CC (I.D. 20)
> 1-877-794-7322 (I.D. 20)
>
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: "Luke Herbert" <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 00:55:35 -0000
Subject: Re: Mersenne: Want to Increase Sales 100 - 150% ?
>What's this garbage doing on the list, apparently directly from someone on
>base.com???
>[Insert rubbish here] (no i am not going to post it again).
Well yes its crap so why did you post it yest again??
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Chris Nash <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 17:19:10 -0500
Subject: Re: Mersenne: Meganet's Primality Code
Hi folks,
Just thought I'd add my 2c worth to the discussion.
Firstly, bashing Professor Milstein is not getting anybody anywhere. His
"endorsement" is merely part of Meganet's commercial blurb, and while some
comments outside the body of the letter are rather eye-catching, isn't that
what advertising is about? As for the letter itself, it doesn't make any
indefensible claims, it doesn't claim a proof, it analyses the method for
its algorithmic qualities.
I'd disagree with some of the claims of efficiency, in that existing
implementations of probable-prime tests return the "same" results in shorter
time. At least, the "same" primes are returned, in shorter time. (I tested
4^7057-3 for probable primality some weeks ago in much less than 33 minutes
on a mere P-200). Pomerance et al tell us that pseudoprimes are rare, but
they also tell us that it's possible to construct counterexamples for any
number of pseudoprime tests. Meganet would therefore only have a
"breakthrough" (either algorithmic or mathematical) if their test was "more
effective", ie was indeed deterministic. Hence we require proof or
counterexample. Either would only be feasibly attained with more details of
the algorithm. And again, don't curse Meganet for being very clandestine
about these details - they are after all using it as a commercial
proposition. That's business.
Saul Backal's comment to Scott Kurowski that the test can degenerate into LL
is really the only lead anyone is going to get on attempts for
counterexamples - it might be worth considering Lucas pseudoprimes (to one
or more discriminants), somewhat tougher a condition than strong
pseudoprimality but still capable of construction. Even *if* the method is
equivalent to a Lucas test, you could be spending a lot of time on the
attempt. As Jud McCraine points out, Maple has been doing this for a long
time - and even if you know how Maple does the test, you may be hard-pressed
to find a counter-example. (the SPRP and LL test question is still
unanswered as far as I know but there seems no reason to believe there is
*no* counter-example - deterministic arbitrary prime polynomial-time testing
would say a lot about GRH).
On the one hand, I'd like to support any efforts to find a counter-example.
On the other hand, I feel maybe it is best to let things be. If Meganet's
target market is "industrial-strength" primes, that's fine. Since however
they addressed us as mathematicians in their initial mailings, I'm imagining
they seek a mathematical market too, unfortunately a market that requires
*proof*. Caveat emptor - there is no top 5000 for "probable primes".
Chris Nash
Lexington KY
UNITED STATES
=======================================================
Co-discoverer of the 7th and 10th largest known primes.
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Stephen Whitis <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 18:39:21 -0600
Subject: Re: Mersenne:Garbage
At 02:43 PM 3/16/99 -0600, you wrote:
>They will probably use your call to confirm that you are receiving the spam,
>and may attempt to obtain further demographic information over the phone.
>They can then market your email address to other spammers for a good price.
That is correct. Following spammers "remove" instructions almost always
results in more spam, not less, because they use it to verify your email
address is valid.
In the case of the spam sent to the Mersenne list, it's the address of the list
itself that would need to be removed. More effective will be to take the
account
which sent it to the list and block it. The mailing list shouldn't allow
anyone
but members of the list to post, so killing their membership to the list should
keep them from continuing.
>DejaNews has a service which provides an email address for use with USENET
>groups.
It's a web-based email address, and can be used for any purpose, not just for
Usenet. I used one for awhile, and now, months later, it still receives spam
emails almost every day. They may filter or block some spam, but not much.
Jason posted...
>The original spam came from a uunet IP address, and I complained to
>[EMAIL PROTECTED] the day I got the ad.
UU.net is probably the biggest provider of spam services online today. Almost
every day, in the NANAE newsgroup, there are complaints about UUnet spam.
Sending them a complaint *might* help, but I wouldn't count on it.
- ---
Stephen Whitis
Visit http://www.whitis.com for information
about Delphi, NT4.0 software, and more.
Support the anti-Spam amendment
Join at http://www.cauce.org/
I won't do business with spammers, but I will report them, ridicule
them, etc. Spam me at your own risk.
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: "Vincent J. Mooney Jr." <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 21:09:29 -0500 (EST)
Subject: Re: Mersenne: Meganet's Primality Code
I asked for Jaime Milstein's Ciriculum Vita, his bio, a background. That is
not bashing.
Others have asked for similar information. We all assumed, I feel and at
least I did, that if there were good solid information, we could better
evaluate his endorsement.
And we do all want to know how good this all is. People have inquiring minds.
At 05:19 PM 3/16/99 -0500, Chris Nash wrote:
>Hi folks,
>
>Just thought I'd add my 2c worth to the discussion.
>
>Firstly, bashing Professor Milstein is not getting anybody anywhere. His
>"endorsement" is merely part of Meganet's commercial blurb, and while some
>comments outside the body of the letter are rather eye-catching, isn't that
>what advertising is about? As for the letter itself, it doesn't make any
>indefensible claims, it doesn't claim a proof, it analyses the method for
>its algorithmic qualities.
>
>snip...snip
>
>Chris Nash
>Lexington KY
>UNITED STATES
>=======================================================
>Co-discoverer of the 7th and 10th largest known primes.
>
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: mersenne-invalid-reply-address
Date: Wed, 17 Mar 1999 06:35:58 +0100
Subject: Mersenne: --= Free Software Club =--
Paying To Much For Overpriced Software????
Would you like to stop paying retail prices for software??? Even on name
brand software like Microsoft, Corel, McAfee, Adobe, Lotus, Claris and
much more. Saving as much as 75% off retail, that means wholesale
pricing to you.
Well now you can !!!!
Join our FREE weekly e-mail subscription, called " SOFTWARE SPECIALS".
And as a bonus, you'll automatically be entered in a drawing each month
for a software package value at over $300. The drawing takes place the 15th
each month. Thereafter you will automatically be re-entered as long as you
are an active subscriber.
For more details and to subscribe click on the link below... NOW
http://3634122867
PLUS, you will also receive the special secret members only URL
For FREE !!! This secret URL will allow you to participate in .......
A Incredible Members Software Special .......
Corel WordPerfect Suite 8, (Full Version CD-Rom, Retail Boxed with
Manual) Priced at an incredible $139.95 !!! Save BIG on this member
special... that's $120 dollars off discount catalog pricing.
You can find this special only by being a subscriber, you can
subscribe by clicking here Now
http://3634122867
The low prices we offer are made possible thanks to our strong buying
capacity, low overhead and small margins.
All prices valid while supply last only. Prices subject to change
without notice.
Subscribe Now For FREE at http://3634122867
Or you can also call for info on how to receive your FREE
Membership and register in the drawing to win $300 worth of
software by calling 800-600-0343 Ext. 1552 Toll Free
_______________________________________________________
We adhere to RESPONSIBLE Email Ethics if you wish to be
"removed" from future mailings, please click below
mailto:[EMAIL PROTECTED]?subject=remove and hit send.
Our software will automatically block you from future mailings.
Or call 800-600-0343 Ext. 1746 and ask to be on our remove list
________________________________________________________
Sender Results Marketing
Address: 800 E. Fairview Rd. Suite 146
Asheville, NC 28803
Phone : 800-600-0343 Ext. 1552
=================================================
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Luke Welsh <[EMAIL PROTECTED]>
Date: Tue, 16 Mar 1999 22:51:41 -0800
Subject: Mersenne: Spam
Hi All--
As most of you know, Majordomo has always been configured to
bounce posts from people who are not subscribed to the list.
In the past, this has caught all the spam (and I have saved it
all, anybody want copies?) Well, one spam did get through
at the dawn of the Age Of Spam.
Today, there was a sudden flurry of subscribes (to both mersenne
and mersenne-digest) from addresses in the .mx domain. We quickly
unsubscribed them all. We also looked for similar new subscribers
over the last few days, and unsubscribed them.
Obviously, one has snuck in. We'll smoke him out. In the meantime,
I suppose a few more spams might be headed our way.
I don't think anything further need to be said on the matter.
- --Luke
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Conrad Curry <[EMAIL PROTECTED]>
Date: Wed, 17 Mar 1999 01:01:54 -0600 (EST)
Subject: Re: Mersenne: LL testing
> I'm curious, as to the nature of the proof that the LL test can definitively
> prove/disprove the primality of a Mersenne number. The resources I have are
> limited and don't go into depth. Before I search elsewhere, I was wondering if
> anyone on the list could help me:
Go to Luke's Mersenne bibliography page at
http://www.scruznet.com/~luke/biblio.htm
You will find an article by Edouard Lucas at
http://www.scruznet.com/~luke/lit/lit_068s.htm
and Lehmer's article "On Lucas's Test for the Primality of Mersenne's
numbers" at http://www.scruznet.com/~luke/lit/lit_007s.htm
> What were the original details of Lucas' test? And how did Lehmer modify it
> into its current form (at least I know how to perform the LL test in its
> current state).
In summary Lucas gave two theorems for 2^(4k+/-1)-1. For N = 2^(4q+1)-1
use the familiar sequence 4, 14, 194, .... For N = 2^(4q+3)-1 use the
sequence 3, 7, 47, ..., which is correct. See R.E.Powers announcements of
primality of M89 (Lucas incorrectly gave it as composite) at
http://www.scruznet.com/~luke/lit/lit_002.txt and for M107 at
http://www.scruznet.com/~luke/lit/lit_003s.htm. For M89 Powers uses the
sequence 4, 14, 194, ... and for M107 he uses 3, 7, 47, ....
In Lehmer's paper he states that the sequence 4, 14, 194, ... can be
used for both cases. There is some question if Lucas' proof is complete,
Lehmer gives both a sufficient and necessary proof of the Lucas test.
> In the LL test, we start with S(1) = 4. The Prime Page says we can use S(1) =
> 10 and certain other values depending on p. Can anyone clarify this?
For starting values we can use 4, 52, 724, ..., U{n}, ..., where U{n} =
14 U{n-1} - u{n-2} and 10, 970, 95050, ..., V{n}, ..., where V{n} = 98
V{n-1} - V{n-2}. Also there are 2^(p-2) starting values that depend upon
p. For example to test 2^5-1 we could use the starting values 4, 9, 10,
11, 20, 21, 22, or 27.
Also see my post to the list in the (defunct) mail archives
http://www2.netdoor.com/~acurry/mersenne/archive3/0679.html
> To prove that M(127) (Or M127, whichever refers to 2^127 - 1, not the 127th
> Mersenne prime) is prime,
M(p) is usually the pth Mersenne number and Mp is 2^p-1 in the
literature. Though occasionally M(p) is used as 2^p-1 on the list. It
could cause confusion only for small p. Is M(3) 2^3-1 or 2^5-1?
> did Lucas use his test by hand? I know he did it by
> hand, at the very least.
I don't know about Lucas. Read some of the articles in Luke's
bibliography, it is a wonderful history. Lehmer and Uhler used desk
calculators. Uhler made many calculations of logarithms and powers,
see for example http://www.scruznet.com/~luke/lit/lit_019.htm
Though poor Uhler had been doing LL-tests in the gap between M127 and
M521.
> I posted a message on sci.math a while ago. The response was deafening. If
> anyone on the list would like to add to the voluminous (har har har) response
> I've received, I'd be grateful. It can be found at:
> http://www.dejanews.com/getdoc.xp?AN=450576076
> I don't wish to add to the size of the list digest unduly.
STL137 asks in his post if there is a test similiar to the LL test for
numbers of the form 2^N-k where k=3,5,7,.... A primality test of the
Lucasian type depends on the factorization of N+1, so I guess not.
However, for some k*2^N-1 there is a primality test that uses the familiar
S{n} = S{n-1}^2-2 differing only in the starting value S{0}. See
Riesel, "Prime Numbers and Computer Methods of Factorization" for example.
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: [EMAIL PROTECTED]
Date: Tue, 16 Mar 1999 23:47:57 -0800 (PST)
Subject: Re: Mersenne:Garbage
<OFF TOPIC>
As a system administrator, I once called one of those numbers
to nicely ask that they pass on to the spammers that we are Washington
residents and that we request that they not spam us any more. A nice lady
answered the phone saying someting or another answering service. I asked
her if I could ask a question and she absolutely flipped! Seems that she
was getting tired of being yelled at. Took me 5 minutes to talk her down.
I felt like I was talking someone off the edge of the side of a building.
Moral: If you call these people, please understand that they are being
yelled at for hours on end. Do try to be nice, you'll get much further!!
Also, they have to take your e-mail address down and place it on their
"don't spam list". Instead of giving them your actual e-mail address, tell
them you are an admin for your domain and that you request they no longer
e-mail to that domain. They can't confirm any one e-mail address that way
and it provides a bit of protection for the rest... blah blah blah take it
as you will...
</OFF TOPIC>
On Mon, 15 Mar 1999, Jonathan A Zylstra wrote:
>
> -this was in the ad, so this might help
>
> >----------------------------------------------------------
> >To be removed, simply call 1-800-600-0343 ext. 1746
> >and leave your e-mail address. Thank you.
> >----------------------------------------------------------
> >>>>>CONTACT INFORMATION<<<<<
>
> Call us TOLL FREE today 1-877-SWIPE-CC (I.D. 20)
> 1-877-794-7322 (I.D. 20)
>
> J. Zylstra
> [EMAIL PROTECTED]
>
>
>
>
> ___________________________________________________________________
> You don't need to buy Internet access to use free Internet e-mail.
> Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
> or call Juno at (800) 654-JUNO [654-5866]
> ________________________________________________________________
> Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
>
--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
: WWW: http://www.silverlink.net/poke : Boycott Microsot :
: E-Mail: [EMAIL PROTECTED] : http://www.vcnet.com/bms :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
From: Conrad Curry <[EMAIL PROTECTED]>
Date: Wed, 17 Mar 1999 01:48:56 -0600 (EST)
Subject: Re: Mersenne: Chance of a Mersenne prime
>
> > Since I joined the project 10 months ago, we have found no
> >new Mersenne primes
>
> We have been unlucky.
If they were easier to find we wouldn't be looking for them.
Since M(5) appeared in a codex in 1456 there have been 32 more Mersenne
primes discovered up to 1998. An average of one every 16.4 years.
If we only consider those found in the computer age when Robinson used
the SWAC to find M(13)-M(17) in 1952, there is a total of 25 between 1952
and 1998. An average of one every 1.84 years. GIMPS has found 3
Mersenne primes in 3 years. I would say we are doing pretty good.
Regarding Noll's island theory, now that p has grown large is it
possible the pattern (756839, 859433), (1257787, 1398269), (2976221,
3021377) continues (ie groups of two separated by gaps)? This would
suggest the next Mersenne prime would be found after a gap perhaps p>6M.
For more see the archives http://www2.netdoor.com/~acurry/mersenne/
Search for "Noll* island".
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
------------------------------
End of Mersenne Digest V1 #533
******************************
