mersenneforum.org

mersenneforum.org (https://www.mersenneforum.org/index.php)
-   News (https://www.mersenneforum.org/forumdisplay.php?f=151)
-   -   41st Known Mersenne Prime Reported!! (https://www.mersenneforum.org/showthread.php?t=2475)

dave_0273 2004-05-15 13:13

41st Known Mersenne Prime Reported!!
 
Ok, trying not to hyperventalate here

[CODE] ------- Mersenne Exponent Test State -------

Assigned in Tests Cleared Since Last Synchronization

Factoring only : 8374 Factored composite : 13128
Lucas-Lehmer testing : 62619 Lucas-Lehmer composite: 74012
Double-checking LL : 14119 Double-checked LL : 39346
Prime, VERIFIED : 1
Prime, UNVERIFIED : 1
---------------------- ------- ---------------------- -------
TOTAL : 85112 TOTAL : 126488

[/CODE]

Has anyone else seen this???

Is this M41??? Or am reading this wrong??

Someone, can you either confirm or reject this???

George??

PrimeCruncher 2004-05-15 14:49

GRRRR! Does somebody have the appropriate data files so I can mine them for information? I KNEW this was going to happen.

tom11784 2004-05-15 14:50

It means someone returned a result stating prime. In two to three weeks or so we'll find out through independent verifications if this is indeed prime, and if so whether it is above or below 20,996,011 meaning 20,996,011 could even be M41.

Only time will tell....


EDIT for Prime's q AGAIN within a min of my post:
Are you talking about status.txt files for now and a day or so ago to see what exponents have had results returned?

PrimeCruncher 2004-05-15 14:57

[QUOTE=tom11784]EDIT for Prime's q AGAIN within a min of my post:
Are you talking about status.txt files for now and a day or so ago to see what exponents have had results returned?[/QUOTE]

I'm talking status and cleared for the hour before it appeared on the summary and the hour it appeared. If anyone has them, please PM me and I'll give you my e-mail address.

dave_0273 2004-05-15 15:09

I don't know exactly when it appeared but it was there in the 13:00 "UTC Time" summary. I have the status and cleared files for that hour.

I too am looking for the exact hour which it appeared. Please PM me if you have those files.

Primecruncher:- What did you mean about the "I KNEW this was going to happen"

PrimeCruncher 2004-05-15 15:18

No, I am not the discoverer of the prime and I am not omniscient. I wish I were, then I wouldn't have to go digging through data files!

What I meant was that a while back, Xyzzy set up the Opteron so that it would make a copy of each data file every hour. Things have changed and George is now in posession of the Opteron and the archiving isn't available. I just KNEW we'd find another prime and I STILL wouldn't have access to the data files!

dave_0273 2004-05-15 15:24

I can just see it now.... I don't think that I am going to get any sleep until this verification run is complete.

I don't have any basis for this at all, but I am about 50% sure that the result is fake.

However, if it is genuine, my guess is that it is a double check around the 10.5 - 11.5M mark. I can't wait to get my hands on the data files to see what the possibilities are.

Prime95 2004-05-15 15:53

I've downloaded the logs - the server did not email me automatically for some reason. So far the result looks genuine. It was run using version 23 by a top 1000 contributor. If proven correct it will be another world record for GIMPS!!

I've emailed the user for his last save file. That will "definitively" prove the report genuine or bogus. I've also emailed Ernst and Guillermo for CPU time on an official verification run.

Keep your fingers crossed!!

PrimeCruncher 2004-05-15 16:02

[QUOTE=Prime95]I've downloaded the logs - the server did not email me automatically for some reason. So far the result looks genuine. It was run using version 23 by a top 1000 contributor. If proven correct it will be another world record for GIMPS!![/QUOTE]

Top 1000 by GIMPS or PrimeNet? By "world record" do you mean bigger than 20996011?

PrimeCruncher 2004-05-15 16:38

Okay, people. Based on Prime95's hints, I've performed an analysis based on all exponents returned 5/15/04 BEFORE 1300 UTC. After eliminating all exponents returned by those not in PrimeNet's top 1000, we are left with 73 possibilities.

Also, the exponent, if prime, is NOT bigger than the current M40. No possibilities are over 20M.

davar55 2004-05-15 17:13

While we're waiting for the EXCITING confirmation:

Has anyone seen THIS conjecture, which is numerically
suggested by and consistent with a simple calculation
with M1 thru "M40": (Check it out)

LIM(n-->infinity) Mn^(1/n) == 3/2

i.e. Mn approx. == K * (3/2)^n for some constant K
i.e. Mn == O((3/2)^N) == BIG-O-OF (3/2)^n

(Possibly) useful for predicting density, if not location,
of MPs, eh?
And the numerical evidence, while brief, is convincing.

PrimeCruncher 2004-05-15 17:23

After further analysis, the field of suspects has been narrowed to 55. Only two possibilities are below M40 but above M39; all of the others are under M39.

Now let's see if I can get some more info from George. :wink:

Two questions:
1. Is the discoverer a team?
2. Is it a first-time test or a double-check?

aaronl 2004-05-15 21:39

I don't see why everyone expects the number to be so small. From the email announcement: "If proven correct, this will be the new largest known prime."

PrimeCruncher 2004-05-15 21:46

[QUOTE=aaronl]I don't see why everyone expects the number to be so small. From the email announcement: "If proven correct, this will be the new largest known prime."[/QUOTE]

What e-mail announcement? Based on cleared.txt from 1500 UTC on 5/15/04 no exponents larger than M40 have been returned since the day began (based on PrimeNet's UTC time).

aaronl 2004-05-15 21:50

[quote]
From: George Woltman <woltman@alum.mit.edu>
Date: Sat, 15 May 2004 11:42:54 -0400
To: [email]prime@hogranch.com[/email]
Subject: [Prime] 41st Mersenne Prime reported!
X-Mailer: QUALCOMM Windows Eudora Version 5.2.1

Hi all,

A computer has just reported discovering the 41st known Mersenne Prime!!
If proven correct, this will be the new largest known prime.

The test was run using version 23 with its better false-positive screening.
I've emailed the user and asked for a copy of the save file with a few
thousand iterations left to go. If he emails that file to me and re-running
the last iterations again shows the number to be prime, then we will
know the report is accurate.

I'll keep you posted,
George
[/quote]

PrimeCruncher 2004-05-15 22:10

:furious: :censored: It seems Excel has a problem with large text imports. In any case, I'll be redoing my analysis and posing here again soon.

PrimeCruncher 2004-05-15 22:31

A-ha! I KNEW I was off the first time! Seventy-THREE possibilities. After the inital round of cuts we now have (drumroll please) seventy-FOUR possibilities. (Notice how that's an inverted 47?)

PrimeCruncher 2004-05-15 23:06

Based on further analysis of program versions in use on each of the possible computers, I've narrowed the list of suspects, once again, to 55. Of the remaining 55, 13 are 33M exponents. So, George, it wouldn't hurt to tell us whether the exponent qualifies for the EFF prize or not. :wink:

Now, does anyone keep hourly copies of summary.txt so we can find out what hour the unverified appeared in?

dave_0273 2004-05-15 23:29

How many people received the email?? I didn't recieve an email!!

flava 2004-05-15 23:38

Hmmm... based on the information available I narrowed down the info from cleared.txt to less than 20 exponents. One of them looks like a probable candidate to me ... and it isn't small. Here is a encrypted prediction: Ír½Mü¶ÞKøBCnXXQñ

Xyzzy 2004-05-15 23:41

Are you subscribed to the mailing list?

S80780 2004-05-15 23:41

[QUOTE=dave_0273]How many people received the email?? I didn't recieve an email!![/QUOTE]
To receive these, you have to subscribe to the Mersenne Mailing List at [url="http://hogranch.com/mailman/listinfo/prime"]http://hogranch.com/mailman/listinfo/prime[/url].
There, you'll also find a link to the archive.

Benjamin

PrimeCruncher 2004-05-15 23:48

[QUOTE=flava]Hmmm... based on the information available I narrowed down the info from cleared.txt to less than 20 exponents. One of them looks like a probable candidate to me ... and it isn't small. Here is a encrypted prediction: Ír½Mü¶ÞKøBCnXXQñ[/QUOTE]

First, how did you get down to 20? Do you know the hour the UNVERIFIED first appeared in?

I have a couple that look good, one especially so also. It's residue is remarkably similar to previous known false residues and, checking LUCAS_V.txt, it appears in a handful of exponents there, and in only one other exponent in cleared.txt.

EDIT: The residue also has two inverted 47s in it, which makes me even more optimistic. :grin:

flava 2004-05-16 00:03

[QUOTE=PrimeCruncher]First, how did you get down to 20? Do you know the hour the UNVERIFIED first appeared in?

I have a couple that look good, one especially so also. It's residue is remarkably similar to previous known false residues and, checking LUCAS_V.txt, it appears in a handful of exponents there, and in only one other exponent in cleared.txt.

EDIT: The residue also has two inverted 47s in it, which makes me even more optimistic. :grin:[/QUOTE]

I don't know the hour but tried to make a reasonable guess. With the number of prime junkies :redface: checking PrimeNet, I don't think it takes too long before the news gets out. :bounce:
I'm not 100% sure I got the right one, of course.
I think I see the one that looks good to you. Mine happens a bit later.

PrimeCruncher 2004-05-16 01:27

Okay, my enciphered prediction:
BFWICH-MJW OZIRZPI, BMQ YCIVAQV QNGBH-JLIP BYBCASMV, CYLPW YKMVQWV JLIPBH-AWDPM

Xyzzy 2004-05-16 02:34

My guess:

[pre]f24d6de6e5087caa136135bd50eff38b[/pre]

Prime95 2004-05-16 03:17

Thank you all for making your predictions encrypted. It adds to the fun and anticipation if we don't prematurely blurt out the new prime.

BTW, it is a new prime. The user sent in his save file. I reran the final 15,000 iterations and prime95 rediscovered the prime. There is no known method for creating a bogus save file that falsely rediscovers the prime.

Finally, does anyone have some big UNIX iron to run an official verification on for a month? Guillermo Ballester only has an Athlon machine to run Glucas on. I haven't heard from Ernst Mayer yet.

justinsane 2004-05-16 04:20

[QUOTE=Prime95]Thank you all for making your predictions encrypted. It adds to the fun and anticipation if we don't prematurely blurt out the new prime.

BTW, it is a new prime. The user sent in his save file. I reran the final 15,000 iterations and prime95 rediscovered the prime. There is no known method for creating a bogus save file that falsely rediscovers the prime.

Finally, does anyone have some big UNIX iron to run an official verification on for a month? Guillermo Ballester only has an Athlon machine to run Glucas on. I haven't heard from Ernst Mayer yet.[/QUOTE]


I'd gladly run an official verification on a UNIX P4 2.4G

[email]vanklein@buffalo.edu[/email]

roy1942 2004-05-16 04:56

In the hour before the 'unverified' notice first appeared, there were 11 exponents checked in that satisfied 'new record', i.e., larger than 20996011. Six of these are from participants ranking in the top 1000. Two of these are 33M exponents.

I don't see how to narrow it down any further.

roy1942 2004-05-16 05:30

[QUOTE=Prime95]Finally, does anyone have some big UNIX iron to run an official verification on for a month? Guillermo Ballester only has an Athlon machine to run Glucas on. I haven't heard from Ernst Mayer yet.[/QUOTE]

A 33M exponent would take about a month using Prime95 on the fastest P4. I don't know how fast "big UNIX iron" running Glucas could do it. Does this request for a month for a verification point to a 33M exponent, or rule it out?

One of the 33M candidates alluded to in my previous post had been running for over a year. If that's the one, it proves the value of persistence!

PrimeCruncher 2004-05-16 12:34

[QUOTE=roy1942]In the hour before the 'unverified' notice first appeared, there were 11 exponents checked in that satisfied 'new record', i.e., larger than 20996011. Six of these are from participants ranking in the top 1000. Two of these are 33M exponents.

I don't see how to narrow it down any further.[/QUOTE]

What hour did the UNVERIFIED appear in?

EDIT: Never mind, I figured it out based on my chart. My guess is one of the six! :banana:

Prime95 2004-05-16 14:17

[QUOTE=justinsane]I'd gladly run an official verification on a UNIX P4 2.4G[/QUOTE]

The ideal machine would be a multi-processor Itanium or Alpha or Sun machine. Glucas is multi-threaded. Guillermo is privately emailing some GLUCAS users to see if any are interested.

PrimeCruncher 2004-05-16 14:19

[QUOTE=Prime95]The ideal machine would be a multi-processor Itanium or Alpha or Sun machine. Glucas is multi-threaded. Guillermo is privately emailing some GLUCAS users to see if any are interested.[/QUOTE]

A month on a multi-processor Itanium? I guess that would point to a 33M exponent...

Prime95 2004-05-16 14:20

[QUOTE=roy1942]Does this request for a month for a verification point to a 33M exponent, or rule it out?[/QUOTE]

I've given fewer hints than in the past because you readers are so good at sleuthing! However, the 10 million digit question is important to many folks. So......the new Mersenne prime is NOT ten million digits.

PrimeCruncher 2004-05-16 14:40

[QUOTE=Prime95]I've given fewer hints than in the past because you readers are so good at sleuthing! However, the 10 million digit question is important to many folks. So......the new Mersenne prime is NOT ten million digits.[/QUOTE]

Well, thanks for the compliment! Well, my guess isn't a 10 million digit number, so I now have a 25% chance of being correct! :banana:

TTn 2004-05-16 15:46

[QUOTE]I've given fewer hints than in the past because you readers are so good at sleuthing! However, the 10 million digit question is important to many folks. So......the new Mersenne prime is NOT ten million digits.[/QUOTE]

George, is that a trick answer?
Is it bigger than 10m digits?
:unsure:

Prime95 2004-05-16 16:23

[QUOTE=TTn]George, is that a trick answer?
Is it bigger than 10m digits?
:unsure:[/QUOTE]

OK, the new Mersenne prime is not ten million or more digits. The EFF award is still up for grabs.

TTn 2004-05-16 16:29

Thanks George!
My plans for today, revolved around that answer.

ixfd64 2004-05-16 18:39

Is it bigger than M20996011?

edit: some say it is, some say it isn't

nitro 2004-05-16 18:43

[QUOTE=S80780]To receive these, you have to subscribe to the Mersenne Mailing List at [url="http://hogranch.com/mailman/listinfo/prime"]http://hogranch.com/mailman/listinfo/prime[/url].
There, you'll also find a link to the archive.

Benjamin[/QUOTE]

Perhaps he means the email that specifies EXACTLY what the new exonent is....oh wait, I forgot it only goes to those of us who have already found a prime :whistle:

nitro 2004-05-16 18:44

[QUOTE=PrimeCruncher]GRRRR! Does somebody have the appropriate data files so I can mine them for information? I KNEW this was going to happen.[/QUOTE]


Haven't you learnt anything from the last time? You know the FALSE positive?? Just leave it until the official announcement.

want to know what it is early, then just do what I did, find a Mersenne prime....

PrimeCruncher 2004-05-16 18:50

[QUOTE=ixfd64]Is it bigger than M20996011?

edit: some say it is, some say it isn't[/QUOTE]

George has said that is is bigger than M40.

As for you, Nitro, a friend I'm talking to on IM who peruses the boards says you're "slightly obnoxious" and I have to agree. Quite frankly, I'll do what I want and since George says this one's the real thing and I think I have the right exponent it looks like my time was well-spent.

clowns789 2004-05-16 19:11

If it is prime it might be in Newsweek. Last time, the finder of M40 had a quote on the quote page.

nitro 2004-05-16 19:35

[QUOTE=PrimeCruncher]George has said that is is bigger than M40.

As for you, Nitro, a friend I'm talking to on IM who peruses the boards says you're "slightly obnoxious" and I have to agree. Quite frankly, I'll do what I want and since George says this one's the real thing and I think I have the right exponent it looks like my time was well-spent.[/QUOTE]

To be perfectly honest I don't give a flying f*** what you think about me. It's my name in the record books not yours.

As to your second point, you started this "I can't wait to work it out" BEFORE George made any confirmation or otherwise.

Guess we were just lucky you didn't work out the last "find" and blurt out some premature announcement aren't we.

By the way you can't "do what you want", when you download and use the software you agree to abide by certain conditions. Don't like them, take your time elsewhere, you won't be missed.

PrimeCruncher 2004-05-16 19:58

[QUOTE=nitro]To be perfectly honest I don't give a flying f*** what you think about me. It's my name in the record books not yours.

As to your second point, you started this "I can't wait to work it out" BEFORE George made any confirmation or otherwise.

Guess we were just lucky you didn't work out the last "find" and blurt out some premature announcement aren't we.

By the way you can't "do what you want", when you download and use the software you agree to abide by certain conditions. Don't like them, take your time elsewhere, you won't be missed.[/QUOTE]

First off, why do I care if your name is in a record book? Your computer found a Mersenne prime. Big whoop. You didn't do the calculations and the assignment of exponents is pretty much random. It could've been someone else just as easily.

As to YOUR second point, I like a challenge. And I've been curious to see whether I'm up to the challenge. Looks like I am this go round.

Third point: As if. :rolleyes: I know the procedure around here, that's why the guess is encrypted. Duh.

Fourth: What software? You mean Prime95? I don't see how that factors into a data analysis.

Fifth: for this board and all others: obnoxious attitude not required.

TTn 2004-05-16 23:23

Hey I thought I was the obnoxious one!
Although the you've reminded me about those so called prize obligations.

A. These agreement terms are not legally binding.

B. Users dont always agree to GIMPS terms(decline), but use the software.

C.The source is easily manipulated anyway.
(I could alter a Prime95 version, for the person who has found the alleged prime. ) Do I get 10%? ...

D. How does George already know this is not a false positive, and its the real thing. My guess is that he is confident in the new error check, though I dont recall that being the case before.

Uncwilly 2004-05-16 23:31

My best understanding is that M41 should have an exponent in the 32,xxx,xxx range. This being the result of it being a certain size larger than M40. This would also require a month run on "big iron".

I think PrimeMonster is farming the area that is in the range above that, where M42 should lie. He has been working up there on TF, P-1, and L-L for quite some time.


I think that some fellow named Mike found M41. :grin:

ixfd64 2004-05-17 00:06

OMG! The third Mike in a row!!!

I wonder if I'll discover a Mersenne prime if I change my name to Mike...

PrimeCruncher 2004-05-17 00:13

[QUOTE=ixfd64]OMG! The third Mike in a row!!!

I wonder if I'll discover a Mersenne prime if I change my name to Mike...[/QUOTE]

Well, we don't know that the discoverer is named Mike yet...

Besides, the name probably has to be natural. :wink:

jinydu 2004-05-17 01:09

On the front page of [url]www.mersenne.org:[/url]

41st Known Mersenne Prime Reported!!

--------------------------------------------------------------------------------

On May 15, 2004, a GIMPS computer reported finding the 41st known Mersenne Prime. Additional quick checks have shown this is nearly certainly a genuine report. However, to be rigorous, we must test the discovery on a different computer using different software. This will probably take a month. The actual number will be announced when that test completes.





He does seem confident that this is the real thing.

HINT: It wasn't me who found it. I'm ranked only #1910...

jinydu 2004-05-17 01:27

On the front page of Mathworld ([url]http://mathworld.wolfram.com[/url]):

41st Mersenne Prime Apparently Found
A computer participating in the Great Internet Mersenne Prime Search reported discovering the 41st known Mersenne Prime on the morning of May 15. Additional details will be posted here as they become available.

lycorn 2004-05-17 09:33

[QUOTE=jinydu]

On May 15, 2004, a GIMPS computer reported finding the 41st known Mersenne Prime.[/QUOTE]

Note that it is stated "a GIMPS Computer", not necessarily a computer connected to Primenet (there some people checking out exponents and reporting results manually); so it may even not be in the Primenet reports... :sleep:
BTW, I´m in the Top 1000 :whistle: !

jinydu 2004-05-17 13:06

Aren't exponents reported manually added to the Primenet database?

tom11784 2004-05-17 13:54

IIRC the Perl errors on the manual submissions had started before this report of a potential M41, so I would think that means either it was sent directly to George, or it was submitted via PrimeNet...

edorajh 2004-05-17 13:58

That's really excellent news!!! It's relatively short time between last finding of Mersenne Prime. Who knows maybe there will be more discoveries this year.

:banana: :banana: :banana:

Lumídek 2004-05-17 14:11

[QUOTE=davar55]While we're waiting for the EXCITING confirmation:

Has anyone seen THIS conjecture, which is numerically
suggested by and consistent with a simple calculation
with M1 thru "M40": (Check it out)

LIM(n-->infinity) Mn^(1/n) == 3/2

i.e. Mn approx. == K * (3/2)^n for some constant K
i.e. Mn == O((3/2)^N) == BIG-O-OF (3/2)^n

(Possibly) useful for predicting density, if not location,
of MPs, eh?
And the numerical evidence, while brief, is convincing.[/QUOTE]

Let me try to find analytically the right value for your conjecture, replacing your guess 3/2. The probability that a large number N is prime is 1 / LN(N). Therefore, the probability that a number comparable to 2^N-1 is prime is therefore naively 1/LN(2^N-1). For large N, neglect 1 and it is 1/LN(2^N) = 1/LN(exp(LN(2).N) = 1/(LN(2).N). Write N as exp(P), and the density of primes in P (the logarithmic scale) becomes 1/(LN(2).exp(P)) times exp(P) = 1/LN(2) which would mean that a new prime appears every time you increase P by LN(2), i.e. if you double M. However, 2^N-1 is guaranteed to be odd, which doubles the probability that it is a prime, and assuming no other correlation or conspiracy, it means that you don't need to double M: it is enough to multiply it by SQRT(2).

So therefore I claim that your limits are true, but the correct number is not 3/2, but rather SQRT(2). Let's call it WhateverIsYourName - Motl's theorem, if no one else knows it. ;-)

Uncwilly 2004-05-17 14:31

You folks may want to look at [URL=http://www.utm.edu/research/primes/notes/faq/NextMersenne.html]http://www.utm.edu/research/primes/notes/faq/NextMersenne.html[/URL]

Quoting from that site: (emphasis mine)
[QUOTE]What about the next Mersenne?
So what can we gather from this? One thing is that given one Mersenne prime exponent's p, the next one will fall, on the average, near [B]1.47576p[/B]. But not too near the average much of the time--sometimes the gaps will be small, sometimes large. So the next (possibly the 41st) Mersenne exponent might be about [B]31,000,000 [/B] yielding a Mersenne with about 9.3 million digits. Or it may not. [/QUOTE]

patrik 2004-05-17 17:48

[QUOTE=tom11784]IIRC the Perl errors on the manual submissions had started before this report of a potential M41, so I would think that means either it was sent directly to George, or it was submitted via PrimeNet...[/QUOTE]
I think it must have been sent to Primenet some way, since we have it in the summary:[CODE] Factoring only : 8312 Factored composite : 13186
Lucas-Lehmer testing : 62446 Lucas-Lehmer composite: 74660
Double-checking LL : 14131 Double-checked LL : 39640
Prime, VERIFIED : 1
Prime, UNVERIFIED : 1
---------------------- ------- ---------------------- -------
TOTAL : 84889 TOTAL : 127488
[/CODE]
Also, George said he didn't get the usual mail from the server.
[QUOTE=Prime95]I've downloaded the logs - the server did not email me automatically for some reason. So far the result looks genuine. It was run using version 23 by a top 1000 contributor. If proven correct it will be another world record for GIMPS!![/QUOTE]

tom11784 2004-05-17 18:41

maybe the reason he didn't get an email has to do with the Perl issues on the server lately

but he found out soon enough anyways and we'll all know in about a month - sooooo far away :bounce:

jinydu 2004-05-18 02:30

Does anyone know where I can find an applet that calculates Li(x) (logarithmic integral, as in the Prime Number Theorem) for VERY large x (as large as the numbers we're testing for primality)?

jinydu 2004-05-18 03:00

[QUOTE=Lumídek]Let me try to find analytically the right value for your conjecture, replacing your guess 3/2. The probability that a large number N is prime is 1 / LN(N). Therefore, the probability that a number comparable to 2^N-1 is prime is therefore naively 1/LN(2^N-1). For large N, neglect 1 and it is 1/LN(2^N) = 1/LN(exp(LN(2).N) = 1/(LN(2).N). Write N as exp(P), and the density of primes in P (the logarithmic scale) becomes 1/(LN(2).exp(P)) times exp(P) = 1/LN(2) which would mean that a new prime appears every time you increase P by LN(2), i.e. if you double M. However, 2^N-1 is guaranteed to be odd, which doubles the probability that it is a prime, and [B]assuming no other correlation or conspiracy[/B], it means that you don't need to double M: it is enough to multiply it by SQRT(2).

So therefore I claim that your limits are true, but the correct number is not 3/2, but rather SQRT(2). Let's call it WhateverIsYourName - Motl's theorem, if no one else knows it. ;-)[/QUOTE]

We can test that assumption. M38 = (2^6972593)-1 is the largest Mersenne prime that we know for sure; i.e. it is known known for sure whether M13466917 is really M39 by size. Also, we have double-checked all the way up to M(8,000,000)

Now, pi(8,000,000) = 539,777
This means that there are 539,777 Mersenne numbers that could be prime, of which only 38 are actually prime. The density is thus about 7.04 * 10^-5.

By comparison Li(M(8,000,000)) / M(8,000,000) ~= 1.80 * 10^-7.

This means that Mersenne numbers are about 391 (not 2) times more likely to be prime than "ordinary" numbers, at least when n < 2^(8 million). However, it should be noted that there is significant error in that estimate because so few Mersenne primes are known (i.e. 38 is a small number, so there is still plenty of room for statistical uncertainty). For instance, the density of Mersenne primes up to M6972592 is significantly different from the density up to M6972593 (i.e. a single Mersenne prime changes the density heavily while a single "ordinary" prime has little effect on the density). Still, I would be quite confident to say that Mersenne numbers less than M(8,000,000) are hundreds of times more likely to prime than ordinary natural numbers less than M(8,000,000)

I think the reason for this is probably because Mersenne numbers have less possible factors than "ordinary" numbers of similar size.

wpolly 2004-05-18 04:15

Could anyone post the 4 candidate exponents along with their residue?

jinydu 2004-05-18 04:15

Just for comparison, I'll make a table.

Let r = (density of Mersenne primes) / (density of regular primes), where (density of Mersenne primes) = (# of Mersenne primes) / (# of prime exponents).

Let n = Mersenne exponent

n-------Mersenne prime density-------Regular prime density----------r
10--------------1.00------------------------0.168----------------5.95
100------------0.400----------------------1.46*10^-2------------27.3
1000---------8.33*10^-2------------------1.44*10^-3------------57.7
10000--------1.79*10^-2------------------1.44*10^-4------------124
10^5---------2.92*10^-3------------------1.44*10^-5------------202
10^6---------4.20*10^-4------------------1.44*10^-6------------291
8*10^6-------7.04*10^-5------------------1.80*10^-7------------390

This table clearly seems to show that Mersenne numbers have an "advantage" over regular numbers, and that this advantage gets larger as the numbers get larger. This means that Mersenne primes do appear to thin out (as everyone probably knows), but not as quickly as regular primes.

In case anyone wants to know, I assumed that Regular prime density = Li (2^n) = (1/ (n*ln 2)) + (1/ (n*ln 2)^2) + (2/ (n*ln 2)^3). I got that formula from [url]http://mathworld.wolfram.com/PrimeNumberTheorem.html[/url].

Double-Checkers: Work hard, and I can add another row to that table!

NOTE: Sorry about the dashes. Its the only way I could make the table display correctly (at least on my browser).

TTn 2004-05-18 06:33

PD versus OD
 
A statistician, would be looking at the least sum of squares. :unsure:
You know that line between predicted data, and observed data. :surprised
This feature is built into RMA, as an accessory!

jinydu 2004-05-18 09:00

[QUOTE=TTn]A statistician, would be looking at the least sum of squares. :unsure:
You know that line between predicted data, and observed data. :surprised
This feature is built into RMA, as an accessory![/QUOTE]

I didn't really have any predicted data. Was just looking at the observed data to look at some general trends. However, a quick check shows that r does NOT grow linearly with exponentially growing n.

EDIT: A quadratic fit works very well.

x = Log (base 10) n
y = r

y = 9.0921x^2 - 6.3465x + 2.0573, with an R^2 value of 0.9994 (according to Microsoft Excel).

If you prefer natural logarithms:

x = ln (n)
y = r

y = 1.7147x^2 - 2.7543x + 2.0516, with an R^2 value of 0.9994

That's a good fit considering the small sample size of only 38 Mersenne primes.

TTn 2004-05-18 09:42

Ah... hem I was refering to:

[url]http://www.utm.edu/research/primes/notes/faq/NextMersenne.html[/url]

The exact line between these lines, providing it is still a candidate ofcourse.
Or more exactly, the line between general Mersenne primes.
(Download RMA for details at:)

[url]http://www.15k.org/rma/[/url]

Shane F.
TTn

jinydu 2004-05-18 10:07

When I look at a graph, the ratio between successive Mersenne prime exponents doesn't seem to be closing in on a limiting value, at least not for the first 38 exponents.

TTn 2004-05-18 10:21

Huh ? Wha?
Did you see the link I gave ya. That's dead on the money.
Well as far as prime numbers are concerned anyway.

Do you mean closing in on a rational limiting value? ie, 3/2
Or irrational limiting value? ie, 1.47576

Well if you meant rational, then ofcourse not.
If you mean irrational, then I suggest you are. :wink: (In a nice manner)

jinydu 2004-05-18 12:01

1 Attachment(s)
Here is the graph I made using Microsoft Excel. The first point is (1, 3/2), the second is (2, 5/3) all the way up to (37, 6972593/3021377). As can be seen, the points don't seem to be closing in on a limiting value towards the right-end of the graph.

AP 2004-05-18 12:16

Verification of M41
 
Hi all,

what about the verification of (probably) M41? Is it already on its way?

Regards
Achim

TTn 2004-05-18 13:00

[QUOTE]jinydu
Here is the graph I made using Microsoft Excel. The first point is (1, 3/2), the second is (2, 5/3) all the way up to (37, 6972593/3021377). As can be seen, the points don't seem to be closing in on a limiting value towards the right-end of the graph.[/QUOTE]

As far as prime numbers are concerned, that's close :ermm:
If you take the average of all, it comes out to be about 1.99.
But take the average of the first 8, or the first 11 Mp, = 1.47... ,

You are right about this though, there is not enough data to do accumulative averages acurately. :no:

Edit: I ran the averages, and it tends to get smoother as n gets large, but when it's rough it really rough. This is due to prime number divergence.

Xyzzy 2004-05-18 13:20

[QUOTE=AP]what about the verification of (probably) M41? Is it already on its way?[/QUOTE]Yes...

Prime95 2004-05-18 18:04

Xyzzy started a P4 verification soon after the prime was found and has completed 5 million iterations.

Ernst started a run using mlucas on a single-cpu system on Sunday. He estimates 3 1/2 weeks to complete the run. He is at 1 million iterations.

Jeff in Canada started a Glucas run Monday on an 8-way Alpha box. He too is at 1 million iterations and matches Ernst's interim residue. If CPU loads stay low Jeff could finish in just over two weeks.

Tony in France also started a Glucas run Monday but ran into some threading problems. His multi-CPU Sun box hasn't reached 1 million iterations yet. However, he going to try and switch to a 16-CPU Itanium machine. No estimates on a possible completion date.

Mike, Ernst, and I will post progress as it comes in. Keep your fingers crossed!

Jeff Gilchrist 2004-05-18 18:21

Its actually a 16-way Alpha box but its shared amongs researchers so I think they are limiting the number of CPUs you can run. I'm actually just using 4 CPUs (that seems to be the max) with GLUCAS but getting pretty good speed.

Lets just hope the queue software is smart enough not to overload the system (ideally it should only let enough processes/threads run for the # of CPUs the box has) so lets hope the load doesn't go up. :bow:

Jeff "in Canada" Gilchrist

PrimeCruncher 2004-05-18 19:24

[QUOTE=wpolly]Could anyone post the 4 candidate exponents along with their residue?[/QUOTE]

Well, GP2 did last time, so I guess I'll do the honors this time. The four remaining possibilities based on the various info posted in this thread:

[CODE]21169397 0x746B874910048A__ curtisc grn16-11
22776563 0xCF3C3B04BD5A10__ curtisc wde3100-028
24036583 0x7EC80125B6DB7E__ jfindley M3
23393483 0xD9BFF123AAE613__ mesirow DellH[/CODE]

Xyzzy 2004-05-18 23:44

6,000,000...

jinydu 2004-05-19 00:05

[QUOTE=PrimeCruncher]Well, GP2 did last time, so I guess I'll do the honors this time. The four remaining possibilities based on the various info posted in this thread:

[CODE]21169397 0x746B874910048A__ curtisc grn16-11
22776563 0xCF3C3B04BD5A10__ curtisc wde3100-028
24036583 0x7EC80125B6DB7E__ jfindley M3
23393483 0xD9BFF123AAE613__ mesirow DellH[/CODE][/QUOTE]

But all of them have a non-zero residue.

Uncwilly 2004-05-19 00:07

[QUOTE=jinydu]But all of them have a non-zero residue.[/QUOTE]

The real number will have a faked residue posted. This keeps it a secret. :rolleyes:

PrimeCruncher 2004-05-19 00:19

[QUOTE=Uncwilly]The real number will have a faked residue posted. This keeps it a secret. :rolleyes:[/QUOTE]

BUT... it is possible to speculate on the identity of the real exponent based on the faked residue in comparison to past fakes. That's how I arrived at my conclusion.

Uncwilly 2004-05-19 00:29

But we were promise better fakes! :rant:

PrimeCruncher 2004-05-19 00:34

It's not as blatantly obvious as last time... but it's still there.

jinydu 2004-05-19 00:41

Hmm, one of those residues does seem a bit suspicious. Why not just randomly generate the entire fake? It would look more convincing.

jinydu 2004-05-19 00:43

[QUOTE=Xyzzy]6,000,000...[/QUOTE]

1 million iterations in 5 hours?! That's good!

PrimeCruncher 2004-05-19 00:46

[QUOTE=jinydu]Hmm, one of those residues does seem a bit suspicious. Why not just randomly generate the entire fake? It would look more convincing.[/QUOTE]

They try... but eventually there are only so many places you can get random values from I guess. Hey, at least it's a little more challenging than last time.

jinydu 2004-05-19 00:49

What was the fake residue last time?

PrimeCruncher 2004-05-19 00:51

Last two fakes:

M39: 0x5A4800136C936D__
M40: 0xB6C80125A48000__

jeff8765 2004-05-19 00:52

It was 0xB6C80125A48000__

EDIT: Prime Cruncher beat me to it. :glare: :razz:

PrimeCruncher 2004-05-19 00:55

[QUOTE=jeff8765]It was 0xB6C80125A48000__

EDIT: Prime Cruncher beat me to it. :glare: :razz:[/QUOTE]

Yup. And I posted the fake residue for M39 too, for comparison. Just so everyone can see how much worse the creation techniques were before...

jinydu 2004-05-19 00:55

Hmm, that casts doubt on my M41 guess...

roy1942 2004-05-19 01:29

[QUOTE=PrimeCruncher]Last two fakes:

M39: 0x5A4800136C936D__
M40: 0xB6C80125A48000__[/QUOTE]

And there's two more fake residues that I have in my notes somewhere, for the two false alarms in the past year. But these were never officially made public.

PrimeCruncher 2004-05-19 01:35

[QUOTE=roy1942]And there's two more fake residues that I have in my notes somewhere, for the two false alarms in the past year. But these were never officially made public.[/QUOTE]

Then how do we know that those are the real fake residues?

Xyzzy 2004-05-19 02:44

[url]http://zdnet.com.com/2100-1104-5215542.html[/url]

roy1942 2004-05-19 02:50

[QUOTE=PrimeCruncher]Then how do we know that those are the real fake residues?[/QUOTE]

-Because some of us figured it out, and not just guessing. The evidence was less hidden in the past, and it was pretty obvious when you had the right one, especially if you had some fortunate archived data files.

What went wrong in those cases was pretty well aired in this forum. It would be up to George or the victims of the glitches to reveal their identity and the exponents.

So far this time, I see no way to choose among the four candidates that satisfy George's hints and the proper reporting hour. I don't see the relevance of the '47' or '74' discussion going on in some messages. If I HAD to guess, I'd choose the lowest one, since I believe the most probable prime is always the lowest exponent if there is no evidence to the contrary. But the differences in this case would be minute.

jinydu 2004-05-19 03:00

M39: 0x5A4800136C936D__
M40: 0xB6C80125A48000__

21169397 0x746B874910048A__ curtisc grn16-11
22776563 0xCF3C3B04BD5A10__ curtisc wde3100-028
24036583 0x7EC80125B6DB7E__ jfindley M3
23393483 0xD9BFF123AAE613__ mesirow DellH

Both of these fake residues have a "4800" in common. No such string appears in any of the four M41 candidates, but M21169397 does have two consecutive zeroes. On the other hand, M23393483 has a "123" and most of the residue follows a "letter, letter, letter, number, number, number" pattern.

KUNCEVO6 2004-05-19 04:03

24036583 has a common sequence with M40

ixfd64 2004-05-19 04:36

[QUOTE=KUNCEVO6]24036583 has a common sequence with M40[/QUOTE]

Yep. Anyone notice the "C80125"?

jinydu 2004-05-19 04:55

And B6 right next to it


All times are UTC. The time now is 20:45.

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.