Mersenne Digest Friday, February 18 2000 Volume 01 : Number 695
----------------------------------------------------------------------
Date: Wed, 16 Feb 2000 22:24:08 +0900
From: "Kotera Hiroshi"<[EMAIL PROTECTED]> (Kotera Hiroshi)
Subject: [none]
Hi all
Is a decimal 23-digit numbers 11111111111111111111111 prime ?
Could you tell me the answer with proof?
24-digit numbers 11111111111111111111 = 101*1100110011001100110011
regards
++++++++++++++++++++++++++++++++++++++++++
$B>.;{!!M5(B
630-8144$B!!F`NI;TEl6e>rD.(B1014-4
phone : 0742-61-8521
email : [EMAIL PROTECTED]
URL : http://www.asahi-net.or.jp/~nj7h-ktr/
++++++++++++++++++++++++++++++++++++++++++
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------------------------------
Date: Wed, 16 Feb 2000 22:27:19 +0900
From: "Kotera Hiroshi"<[EMAIL PROTECTED]> (Kotera Hiroshi)
Subject: Mersenne: 23-digit numbers
Hi all
Is a decimal 23-digit numbers 11111111111111111111111 prime ?
Could you tell me the answer with proof?
24-digit numbers 11111111111111111111 = 101*1100110011001100110011
regards
++++++++++++++++++++++++++++++++++++++++++++
Hiroshi Kotera
1014-4 Tokujyo-cho
Nara City 630-8144 JAPAN
email : [EMAIL PROTECTED]
http://www.asahi-net.or.jp/~nj7h-ktr/
++++++++++++++++++++++++++++++++++++++++++++
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------------------------------
Date: Wed, 16 Feb 2000 08:24:48 -0600
From: "Chris K. Caldwell" <[EMAIL PROTECTED]>
Subject: Mersenne: Re:
At 10:24 PM 2/16/00 +0900, Kotera Hiroshi" (Kotera Hiroshi wrote:
>Is a decimal 23-digit numbers 11111111111111111111111 prime ?
>Could you tell me the answer with proof?
Yes--this is small so you could divide by all integers to the square root.
See http://www.utm.edu/research/primes/prove/ for faster ways.
If you use 317 or 1031 digits it is also a known prime. 49081 '1's gives a
probable-prime. See
http://www.utm.edu/research/primes/glossary/Repunit.html
Chris Caldwell
[EMAIL PROTECTED]
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Date: Wed, 16 Feb 2000 09:45:06 -0600
From: "Robert G. Wilson v" <[EMAIL PROTECTED]>
Subject: Mersenne: Re:
Yes it is. In fact there are five numbers with this
characteristic. (10^n -1)/9 is prime when n is 2, 19, 23, 317 & 1031.
See "The Encyclopedia of Integer Sequences" sequence M2114, Neil J.A.
Sloane and Simon Plouffe, Academic Press, 1995.
Goto: http://www.research.att.com/~njas/sequences/index.html
Sequentially yours,
Robert G. 'Bob' Wilson v,
PhD ATP / CFI
named my the authors as "our most prolific contributor on new
sequences."
"Kotera Hiroshi (Kotera Hiroshi)" wrote:
> Hi all
> Is a decimal 23-digit numbers 11111111111111111111111 prime ?
> Could you tell me the answer with proof?
>
> 24-digit numbers 11111111111111111111 = 101*1100110011001100110011
>
> regards
>
> ++++++++++++++++++++++++++++++++++++++++++
> $B>.;{!!M5(B
> 630-8144$B!!F`NI;TEl6e>rD.(B1014-4
> phone : 0742-61-8521
> email : [EMAIL PROTECTED]
> URL : http://www.asahi-net.or.jp/~nj7h-ktr/
> ++++++++++++++++++++++++++++++++++++++++++
>
> _________________________________________________________________
> Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
> Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
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------------------------------
Date: Wed, 16 Feb 2000 17:23:53 +0100
From: Paul Landon <[EMAIL PROTECTED]>
Subject: Mersenne: Re:23-digit numbers
Hiya Kotera,
(10^23-1)/9 is prime!
[N-1, Brillhart-Lehmer-Selfridge] (digits:23)
Primeform: CHK:CE16CAB
You have correctly spotted patterns there.
(10^24-1)/9 is divisible by at least:-
11
111 = 3.37
1111 = 11.101
111111 = 11.(3.37).91; 91=7.13
11111111 = 11.101.10001; 10001=73.137
111111111111 = 111.1111.900991; 900991=7.13.9901
(10^24-1) / (10^12-1) = 10^12+1
10^12+1 = 73.137.99990001
(10^24-1) / (10^8-1) = 10^16+10^8+1
(10^16+10^8+1)/111 = 90090090990991
90090090990991 = 7.13.9901.99990001
Clearly 111111111111/11 = 10101010101
and 111111111111/111 = 1001001001 etc.
showing that 2^ab-1 is divisible by 2^a-1 and 2^b-1
and can never therefore be prime, and the same for
other bases.
People who know more about this than me, forgive me
for jumping in first, but it is close to my level
of Maths.
Cheers,
Paul Landon
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------------------------------
Date: Wed, 16 Feb 2000 18:46:15 +0000
From: Alexander Kruppa <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Re:
"Robert G. Wilson v" wrote:
> Yes it is. In fact there are five numbers with this
> characteristic. (10^n -1)/9 is prime when n is 2, 19, 23, 317 & 1031.
> See "The Encyclopedia of Integer Sequences" sequence M2114, Neil J.A.
> Sloane and Simon Plouffe, Academic Press, 1995.
>
Are all others proved composite, or are the bigger ones merely beyond the
current limit of primalty testing?
Ciao,
Alex.
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Date: Wed, 16 Feb 2000 11:59:02 -0600
From: "Robert G. Wilson v" <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Re:
n=49081 is a probable prime and the limit of testing today is about 50,000.
Alexander Kruppa wrote:
> "Robert G. Wilson v" wrote:
>
> > Yes it is. In fact there are five numbers with this
> > characteristic. (10^n -1)/9 is prime when n is 2, 19, 23, 317 & 1031.
> > See "The Encyclopedia of Integer Sequences" sequence M2114, Neil J.A.
> > Sloane and Simon Plouffe, Academic Press, 1995.
> >
>
> Are all others proved composite, or are the bigger ones merely beyond the
> current limit of primalty testing?
>
> Ciao,
> Alex.
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Date: Wed, 16 Feb 2000 15:44:31 -0600
From: Ken Kriesel <[EMAIL PROTECTED]>
Subject: Mersenne: Re:
At 10:24 PM 2/16/2000 +0900, "Kotera Hiroshi"<[EMAIL PROTECTED]> wrote:
>Hi all
>Is a decimal 23-digit numbers 11111111111111111111111 prime ?
>Could you tell me the answer with proof?
>
>24-digit numbers 11111111111111111111 = 101*1100110011001100110011
>
>regards
For the smaller numbers get ubasic and run aprt-cle.ub. It shows it is
prime.
Also included with ubasic is ecm.ub and many other programs.
The numbers above are repunits. Repunits with nonprime number of digits
are provably nonprime. (This is why the mersenne search only considers
prime exponents. If a number has n identical digits r in some number base b,
it's a repunit, and if n is factorable into i and j, the number has at
least factors
f and g:
f=sum from k=0 to i-1 of r b^k
and similarly g=sum from k=0 to j-1 of r b^k)
since the nonprime number above has 24 digits, it's divisible by more factors
than you show. By inspection, we can see it's divisible by 3; its digit count
is divisible by 2, 3, 4, 6, and 12, predicting divisors of 11, 111, 1111,
111111,
and 111111111111. Note not all of these are prime either.
Prime Factorization by ECM
Input an integer =? 111111111111111111111111
3 * 7 * 11 * 13 * 37 * 73 * 101 * 137 * 9901 * 99990001
aprt-cle.ub
Finds first factor or indicates primality.
Accepts numbers of form 2^727-1 but apparently not x / y
"APRT-CLE" is the extended version of "APRT-CL"
Cohen-Lenstra version of Adleman-Pomerance-Rumely Test
programmed by Koichiro Akiyama
1988 2 12
modified for UBASIC version 7 by Yuji KIDA
1989 1 31 & 1989 3 30
amended by Frank O'Hara
1990 1 28
extended to 844 digits using extra memories by Yuji KIDA
1991 9 30
extended to 844 digits WITHOUT using extra memories by Frank O'Hara
1991 12 13
rearranged by Yuji KIDA
1992 1 24
Hmm. These numbers are in a sense the decimal analog of mersenne numbers.
I get primes for # of digits n=2,19, 23 and no others below 72,
and first factors for the nonprimes (of prime length) as follows:
n f
3 3
5 41
7 239
11 21649
13 53
17 2071723
29 3191
31 2791
37 2028119
41 83
43 173
47 35121409
53 107
59 2559647034361
61 733
67 493121
71 ?
Other than the special case 3, the factors above are all of form f=2kn+1, like
for mersenne numbers.
Ken
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------------------------------
Date: Wed, 16 Feb 2000 19:00:56 EST
From: [EMAIL PROTECTED]
Subject: Mersenne: Re: those outrrrrageous mathematicians
Paul Landon wrote:
>As an off-topic footnote, anyone interested in History or Biographies
>should look up the some of the flowery characters mentioned above.
>Some biographies have been "ethically cleansed" and have missing data,
>data that should never be allowed to detract from their Mathematical
>and Engineering greatness but is still fascinating. Some of those
>stories could cause tittering schoolboys to find Maths interesting or
>sniggering schoolgirls to enjoy Computing. They could even be used to
>justify increased Federal funding; so that this century's Ada doesn't
>have to finance herself that way ;-)
Paul, are there any biographies (esp. of Ada, Babbage or Turing) which
you especially recommend?
I found 8 matches for the search string "ada lovelace" at amazon.com,
but few of these had accompanying reviews, and for those that did, the
reviews often diverged wildly. The ones that look the most promising:
1) "Ada, the Enchantress of Numbers: A Selection from the Letters of
Lord byron's Daughter and Her Description of the First Computer"
Betty A. Toole (ed.), Ada King Lovelace / hardcover / 1998
Betty A. Toole appears to be an acknowledged expert on Ada Lovelace,
but both of the positive editorial notes are provided by Ms. Toole
herself. The book description doesn't mention whether this is basically
just a collection of letters or whether Toole and Lovelace provide
historical context and interpretation along with the letters. See
also my note about book 2.
2) "Ada, the Enchantress of Numbers: Prophet of the Computer Age"
Betty A. Toole / softcover / 1998
There appears to be a stark division by gender in reviews of this
(and other) books about Ada: the men mostly seem to argue that she
gets way too much credit, and the women idolize her. This book had
3 customer reviews on amazon.com: the first (a man) says it's good
but overly flattering, the second (a woman) says it's excellent,
and the third (a man) calls it hero-worship rather than biography.
3) "The Calculating Passion of Ada Byron"
Joan Baum / hardcover / 1986
There were no editorial descriptions or customer reviews of this book.
Has anyone out there read it?
4) "Ada: A Life and Legacy"
Dorothy Stein / out of print
Anybody read this one?
5) "Ada Byron Lovelace: The Lady and the Computer" (People in Focus
Series) / Mary Dodson Wade / publisher out of stock.
Seems to be geared towards the teen set. Anybody read it?
Cheers,
- -Ernst
p.s.:"Ada, or Ardor" by Nabokov is not about the aforementioned Ada, but
is worth reading nonetheless. :)
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------------------------------
Date: Wed, 16 Feb 2000 20:06:12 -0800
From: Spike Jones <[EMAIL PROTECTED]>
Subject: Mersenne: 1.5 gigahz chip
Well, those two local companies are at it again. The one from
Santa Clara announces a 1.1 GHz chip, then the one from
Sunnyvale announces a 1.5 GHz chip. GIMPS is gonna be
smoking once those two guys start shipping these hummers.
spike
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Date: Thu, 17 Feb 2000 02:05:56 EST
From: [EMAIL PROTECTED]
Subject: Mersenne: Re: Mersenne Digest V1 #694
<< Hi all
Is a decimal 23-digit numbers 11111111111111111111111 prime ?
Could you tell me the answer with proof? >>
Giving proof of primality in a text message? Hrm. Interesting.
S. "That's not even related to Mersennes distantly... sort of" L.
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Date: Thu, 17 Feb 2000 03:16:02 -0800
From: Paul Leyland <[EMAIL PROTECTED]>
Subject: RE: Mersenne: Re: Mersenne Digest V1 #694
> Is a decimal 23-digit numbers 11111111111111111111111 prime ?
> Could you tell me the answer with proof? >>
>
> Giving proof of primality in a text message? Hrm. Interesting.
Ok, here's a proof by converse-of-Fermat:
Let p be the number above. Then p-1 has the prime factorization:
2 * 5 * 11 * 11 * 23 * 4093 * 8779 * 21649 * 513239
All we need to do is find a value "a" such that a^((p-1)/ q) is not
equivalent to 1 modulo p for all of the primes q which divide p - 1.
(Actually, this is stricter than necessary, as a different "a" could be
chosen for each q, but never mind.)
I claim that a=11 solves this problem. The values of 11^((p-1)/q) for each
value of q is:
2 11111111111111111111110
5 5377703061176866466164
11 9819808773394829497336
23 1000
4093 6869680499138125330855
8779 5523680250213453961701
21649 8541468742226406455944
513239 10285654293302278381846
Q.E.D.
To do these computations, I used the GMP calculator at
http://www.swox.com/gmp/index.html
Paul
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Date: Thu, 17 Feb 2000 03:25:04 -0800
From: Paul Leyland <[EMAIL PROTECTED]>
Subject: RE: Mersenne: Re: Mersenne Digest V1 #694
Oops. Minor typo:
Where I wrote:
> All we need to do is find a value "a" such that a^((p-1)/ q) is not
> equivalent to 1 modulo p for all of the primes q which divide p - 1.
> (Actually, this is stricter than necessary, as a different "a" could be
> chosen for each q, but never mind.)
>
> I claim that a=11 solves this problem. The values of 11^((p-1)/q) for
each
I meant:
"The values of 11^((p-1)/q) modulo p for each ..."
which should have been clear from the previous paragraph.
Paul
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Date: Thu, 17 Feb 2000 17:07:19 EST
From: [EMAIL PROTECTED]
Subject: Mersenne: Re: computational history of the Mersenne primes
Many thanks to Paul Landon for the very informative and enjoyable
trilogy about the history of computer testing of Mersenne numbers!
So Newman of Manchester was apparently the first - wait, I just got
an e-mail from someone with the handle [EMAIL PROTECTED] -
must work for some cryptography outfit. No, he goes on to explain that
"...Thanks to ze efforts of some enlightened mediums who are
knowledgeable about zis 'Internetz' of your modern era, selected
of us restless departed souls have been granted limited browsing
und e-mail privileges from ze great beyond. Zese mediums are
automatically transcribing ze voices zey hear, vich means zat
my missives vill probably sound like badly accented Tcherman,
even zo in reality mein written English ist sehr gut."
{Bunch of blah blah about his childhood and career snipped...
can we just cut to the chase here, fella?}
"...ze very first thing I typed into zis 'search maschine' was
ze phrase 'Mersenne prime', und I haff subsequently spent many
months vading sroo mostly tedious messages about 'Mein Komputer
ist so much faster zen yoors' und 'mein Hard drive ist very big',
but occasionally encountering an item of genuine interest, zo
ze accompanying mathematics ist usually ganz falsch. Ze recent
posting by Mr. P. Landon of Lucent Technologie vus very interesting,
but I must state for ze rekord zat I vas avare of zis at ze time.
After all, Newman vas really chust a persona I invented so I could
obtain verk in England. Ze phrase 'Neu Mann' translates into 'New
man' in English. Neumann/Newman - quite clever, nicht wahr?"
At this point this guy's delusions of grandeur just became too much, so
I added .crypt to my spam filter and got back to the truly grave business
of playing Quake.
Seriously, Paul, there's just one quote in your compilation I take issue
with, in the section from Alan Hodges (about the Mersenne primes M521,
M607, M1279, M2203 and M2231 found by Robinson on the SWAC):
> The largest known prime now is again a Mersenne prime, and found
> by exactly the same method, only on a somewhat larger and faster
> computer).
"Exactly the same method" is misleading. Yes, the modern programs all
use the Lucas-Lehmer test in some form. But the algorithmic implementation
thereof has undergone radical changes since the days of Robinson's work.
Most notably, the use of FFT-based multiply schemes have reduced the number
of operations needed to test an n-bit Mersenne number from the
approximately O(n^3) (the n^2 term becomes negligible as n gets large)
cited by Robinson to just O(n^2*log2(n)). The Crandall/Fagin discrete
weighted transform method further eliminates the need for zero-padding
the vectors to be FFTed and more than doubles the speed of a typical
FFT-based implementation. As a result, the constant of proportionality
implied by the O(...) notation is typically around 5, where I've
factored in the fact that each computer input digit in a modern scheme
is around 16 bits or more. If one uses a simple grammar-school multiply
method on a 36-bit architecture like the SWAC, one can have input digits
of similar size (18 bits max.), and needs roughly 2*(n/18)^2 operations
per squaring, and n such squarings means O(n^3) machine operations, with
a constant of proportionality of around 1/100 (call it 1/2^7.)
That means that for a Mersenne of around 8-9 million (call it 2^23) bits
(ropughly the size of current GIMPS first-time tests), these genuine
algorithmic improvements have reduced the labor from around (2^23)^3/2^7
= 2^62 machine operations to test primality (which would need over 100
years even running at 1 Giga-op per second) to around 5*n^2*log2(n)
~= 2^49, a reduction of around a factor of ten thousand.
Luke Welsh informs me that the SWAC had a cycle time of 8 microseconds
(not bad at all for its era!), so the hardware has speeded up since
then by a similar factor of about ten thousand. Thus, algorithmic
and hardware improvements have contributed about equally to increasing
the speed of testing, and thus the phrase "...and found by exactly
the same method, only on a somewhat larger and faster computer" is
(from an algorist's perspective) grossly inaccurate.
So, Paul, perhaps you could add a Chapter 4 to your trilogy, describing
some of the algorithmic and computer advances (and the people who took
adavantage of them to find new Mersenne primes) since the 1950s, as
well as the era of distributed supercomputing which the Internet has
made possible.
Thanks again for the trilogy, Paul!
Cheers,
- -Ernst
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Date: Thu, 17 Feb 2000 16:10:19 -0600
From: "Willmore, David" <[EMAIL PROTECTED]>
Subject: RE: Mersenne: Re: credit for manual testing
How about just "give me 'n' DC/LL/Factoring assignments". Let me worry
about how fast my machines will complete them. You could even let me set
the expiry time (up to the max of 120 days). :)
> -----Original Message-----
> From: [EMAIL PROTECTED] [SMTP:[EMAIL PROTECTED]]
> Sent: Thursday, February 10, 2000 5:11 PM
> To: [EMAIL PROTECTED]
> Subject: Mersenne: Re: credit for manual testing
>
> Bill Rea wrote:
>
> >Do people using the manual check out forms get in the Top Producers
> >list? I ask because I've never been able to find myself in the list
> >and I had an email from a former GIMPS contributor who claimed he
> >got no credit for exponents tested through the manual check out/check in
> >pages.
>
> So far as I know, manual testing work doesn't get credited on the PrimeNet
> top producers pages, but does appear on the GIMPS top producers pages:
>
> http://www.mersenne.org./top.htm <== top 100 based on P90 CPU-years
> http://www.mersenne.org./top2.htm <== everybody else
>
> I don't know if manual testing results are included in the PrimeNet
> total FLOPS throughput figure, which has jumped appreciably in the
> last month.
>
> Thanks to Scott for adding CPU MHz and program version fields to the
> PrimeNet account status pages. One problem i noticed, however, is that
> all of my manual testing machines are listed as running Prime95 v16,
> even the ones that have in the past reported results using Mlucas.
>
> Lastly, it would also be nice to be able to enter something other than
> an x86 CPU type when requesting work from the PrimeNet server - right
> now I'm just clicking on "pentium" for all my non-x86 machines, so as
> to get an appropriate amount of work assigned. There could be preset
> fields for the better-known RISC CPUs, e.g.
>
> Alpha SGI/MIPS Sparc IBM RS6000 Mac/PowerPC
> --------- --------- --------- --------- ----------
> 21064/ev4 R3000 Sparc 1 (not sure ?
> 21164/ev5 R4000 Sparc 2 about the ?
> 21264/ev6 R5000 Sparc 5 early ones) 401
> R8000 Sparc 10 Power 1/SP1 403
> R10000 Ultra 1 Power 2/SP2 G3
> R12000 Ultra 2 Power 3/SP3 G4
>
> and an "other" field, where users can manually enter a type and speed
> for any other CPU types.
>
> I've ignored fine distinctions, e.g. the Alpha ev45 and ev56 and MIPS 4400
> in the above, and some of the types may be redundant, but the table is
> only for purposes of illustration. If Scott decides this is worthwhile,
> he can appeal to the axperts with various platforms for an appropriately
> representative set of CPU types.
>
> Cheers,
> -Ernst
>
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Date: Thu, 17 Feb 2000 18:14:51 -0500
From: "Vincent J. Mooney Jr." <[EMAIL PROTECTED]>
Subject: Mersenne: I get credit
Bill Rea wrote, in answer to a question:
>Do people using the manual check out forms get in the Top Producers
>list? I ask because I've never been able to find myself in the list
>and I had an email from a former GIMPS contributor who claimed he
>got no credit for exponents tested through the manual check out/check in
>pages.
(Bill said) So far as I know, manual testing work doesn't get credited on
the PrimeNet
top producers pages, but does appear on the GIMPS top producers pages:
*******************************
Bill, I do check out manually and I have been getting credit. It seems slow
but George kindly explain he does the updates on a weekend. I have noticed
it can be two weeks before the credit I am due is posted. But I must insist
it is ALWAYS posted (Yes, George, I check on it).
My notes show
992 Vincent J. Mooney 6.18 260 February 13, 2000
(After 9,008,369 was included)
1030 Vincent J. Mooney 5.78 259 January 25, 2000
989 Vincent J. Mooney 5.78 259 January 17, 2000
(after 9,008,309 was included)
1049 Vincent J. Mooney 5.39 258 January 6, 2000
(after 7,298,989 was included)
so I keep track in a simple way. # 1016 today on Top Producers.
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Date: Thu, 17 Feb 2000 17:18:04 -0800
From: Russel Brooks <[EMAIL PROTECTED]>
Subject: Mersenne: Prime95 runs slower and slower over time
I've got another odd, Prime95 related, problem on 1 of the 3 pcs I'm
running GIMPS on. The machine is a 450MHz PII with 256Meg running win98
and nothing else 99.9% of the time. It is doing double checking of
exponent 4777027. It starts off with an iteration time of about .13
seconds. After about 8-12 hours it is still at .13 secs but if left
alone for several days the iteration time climbs to about .39 seconds. I
don't know if it a sudden jump or a gradual change or when it exactly
happens. Rebooting the machine resets the iteration time to the lowest
value.
As an experiment I rebooted the pc and then shutdown Prime95; that was
last Sunday. When the machine was checked last night Prime95 was
started and it immediately started running at the LOW iteration of .13.
This tells me that it isn't something else that is causing the iteration
times to increase; it is Prime95 itself that is having the problems.
Is there anything I should look for?
cheers... Russ
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Date: Fri, 18 Feb 2000 15:37:31 +0100
From: Paul Landon <[EMAIL PROTECTED]>
Subject: Mersenne: Re: computational history of the Mersenne primes
Hiya Ernst,
it wasn't me who wrote "exactly the same method", I was quoting
Alan Hodges, and I found the phrase entertaining too.
[EMAIL PROTECTED] wrote:
[snip New Men, replied separately]
> Seriously, Paul, there's just one quote in your compilation I take issue
> with, in the section from Alan Hodges (about the Mersenne primes M521,
> M607, M1279, M2203 and M2231 found by Robinson on the SWAC):
>
> > The largest known prime now is again a Mersenne prime, and found
> > by exactly the same method, only on a somewhat larger and faster
> > computer).
>
> "Exactly the same method" is misleading. Yes, the modern programs all
> use the Lucas-Lehmer test in some form. But the algorithmic implementation
> thereof has undergone radical changes since the days of Robinson's work.
I agree with you completely!
I did try to hint at this when I wrote:-
> (That has O(n^3), more than George's great implementation of the LL
> test. The Lucas-Lehmer algorithm was created in 1930, but they must have
> used Grammar School multiplication as the Karatsuba method or Fourier
> Transforms had not been invented yet).
>
The simplistic story of mine does leave space for more needed detail,
and you are better than I at describing it.
> Most notably, the use of FFT-based multiply schemes have reduced the number
> of operations needed to test an n-bit Mersenne number from the
> approximately O(n^3) (the n^2 term becomes negligible as n gets large)
> cited by Robinson to just O(n^2*log2(n)). The Crandall/Fagin discrete
> weighted transform method further eliminates the need for zero-padding
> the vectors to be FFTed and more than doubles the speed of a typical
> FFT-based implementation. As a result, the constant of proportionality
> implied by the O(...) notation is typically around 5, where I've
> factored in the fact that each computer input digit in a modern scheme
> is around 16 bits or more. If one uses a simple grammar-school multiply
> method on a 36-bit architecture like the SWAC, one can have input digits
> of similar size (18 bits max.), and needs roughly 2*(n/18)^2 operations
> per squaring, and n such squarings means O(n^3) machine operations, with
> a constant of proportionality of around 1/100 (call it 1/2^7.)
>
> That means that for a Mersenne of around 8-9 million (call it 2^23) bits
> (ropughly the size of current GIMPS first-time tests), these genuine
> algorithmic improvements have reduced the labor from around (2^23)^3/2^7
> = 2^62 machine operations to test primality (which would need over 100
> years even running at 1 Giga-op per second) to around 5*n^2*log2(n)
> ~= 2^49, a reduction of around a factor of ten thousand.
Exactly! I do have a high respect for people who create new algorithms.
This speed up is for free and forever, rather than building a computer
10,000 times faster.
> Luke Welsh informs me that the SWAC had a cycle time of 8 microseconds
> (not bad at all for its era!), so the hardware has speeded up since
> then by a similar factor of about ten thousand. Thus, algorithmic
> and hardware improvements have contributed about equally to increasing
> the speed of testing, and thus the phrase "...and found by exactly
> the same method, only on a somewhat larger and faster computer" is
> (from an algorist's perspective) grossly inaccurate.
I agree. It wasn't me who wrote it!
>
> So, Paul, perhaps you could add a Chapter 4 to your trilogy, describing
> some of the algorithmic and computer advances (and the people who took
> adavantage of them to find new Mersenne primes) since the 1950s, as
> well as the era of distributed supercomputing which the Internet has
> made possible.
Or Chapter 2 Version 0.1.2?
Yes, there is also a 25,000 times speed up from cooperation and distribution.
>
>
> Thanks again for the trilogy, Paul!
>
> Cheers,
> -Ernst
Even funnier, is the following quote from Paul Hoffman's
"The Man Who Loved Only Numbers", a biography of Erdös,
page 35.
"The technique that the GIMPS project uses isn't much
more sophisticated than the 2000 year old method called
"the sieve of Eratosthenes," invented by Eratosthenes of
Alexandria, whose nickname was Beta"
That would be the Beta release of Prime95 then? ;-)
Cheers,
Paul Landon
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End of Mersenne Digest V1 #695
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