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John
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Message 11690 - Posted: 25 Nov 2008 | 22:18:53 UTC

Welcome to the Primorial Prime Search

In order to define a primorial prime, we must first define primorial. The primorial pn# is defined as the product of the first n primes. For example,

p5# = 2*3*5*7*11 = 2310

This can also be notated as p#, the product of all primes less than or equal to prime p. The above p5# would be 11#, the product of all primes less than or equal to prime 11.

Primorial primes are prime numbers of the form p#+/-1. Using the example above, we would look to see if 11#+1 and 11#-1 are prime.

11# = 2*3*5*7*11 = 2310
11#+1 = 2311 is prime
11#-1 = 2309 is prime

Therefore, 11#+1 and 11#-1 are both primorial primes. Using another example, 7#:

7# = 2*3*5*7 = 210
7#+1 = 211 is prime
7#-1 = 209 is not prime

Therefore, only 7#+1 is a primorial prime.

To date, the largest known primorial prime is 392113#+1 with 169966 digits, found in 2001 by Daniel Heuer. 392113#+1 = 2*3*5*7*...*392099*392101*392111*392113 + 1

p#+1 is prime for primes p=2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, and 42209, 145823, 366439 and 392113 (169966 digits).

p#-1 is prime for primes p=3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, and 15877 (6845 digits).

A list of the top 20 primorial primes can be found at The Prime Pages: The Top 20

Mark Rodenkirch's (in collaboration with Geoff Reynolds) psieve program will be used to sieve and Chris Nash and Jim Fougeron's PFGW program will be used to primality test.

Additional information can be found here:

Primorial prime - The Prime Glossary at the Prime Pages
Primorial prime - Wikipedia
Primorial prime - Wolfram MathWorld

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In 2005, a previous effort, "Coordinated Search for Primorial Primes", reached n=637K for p# + 1 and n=650K for p# - 1. Up to those limits, PrimeGrid's Primorial Prime Search will be a doublecheck.

However, there were open reservations up to the following:

n<750K for p# + 1
n<730K for p# - 1

It is unknown how much of these open reservations, if any, were completed.

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

To participate in the sieving effort, see here:
Primorial Prime Search Sieving

To participate in the primality testing effort, see here:
- PRPNet - port 9191

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Message 15109 - Posted: 24 Apr 2009 | 21:16:12 UTC

The wikipedia-link seems to be changed, this is the new one:

http://en.wikipedia.org/wiki/Primorial_prime
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Message 15365 - Posted: 4 May 2009 | 2:24:33 UTC - in response to Message 15109.

The wikipedia-link seems to be changed, this is the new one:

http://en.wikipedia.org/wiki/Primorial_prime

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Message 15367 - Posted: 4 May 2009 | 2:46:40 UTC

Primorial Prime Search enters PRPNet!!!

With the release of PRPNet 2.0.4, we now have the ability to incorporate other prime forms into our search. This upgrade supports the addition of the primorial/factorial prime searches through the use of PFGW.

Therefore, the Primorial Prime Search is now open for primality testing. :) You'll find it on port 9191 in PRPNet.

This prime search is another tedious one in which we're looking for rare and unique primes. However, once found, these primes will remain in the history books for many years to come.

Best of Luck!!!

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Reminder:

In 2005, a previous effort, "Coordinated Search for Primorial Primes", reached n=637K for p# + 1 and n=650K for p# - 1. Up to those limits, PrimeGrid's Primorial Prime Search will be a doublecheck...looking for any missed primes.
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Message 15372 - Posted: 4 May 2009 | 9:43:46 UTC

I've switched couple of cores to Primorial...
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Message 15674 - Posted: 18 May 2009 | 0:23:25 UTC - in response to Message 15372.

I have successfully ported PFGW to MacIntel. This is great if anyone desiring to participate in the Primorial PRPNet project.

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Message 15675 - Posted: 18 May 2009 | 0:29:34 UTC - in response to Message 15674.

I have successfully ported PFGW to MacIntel. This is great if anyone desiring to participate in the Primorial PRPNet project.

That sounds great :)

/Lennart

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Message 17522 - Posted: 18 Aug 2009 | 14:11:38 UTC

[2009-08-18 14:02:21 GMT] PRS: 392113#+1 is Prime!

Is it the largest known primorial prime?

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Message 17524 - Posted: 18 Aug 2009 | 14:29:22 UTC

Yes, it is a known since Sep 2001
http://primes.utm.edu/top20/page.php?id=5
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Message 17526 - Posted: 18 Aug 2009 | 16:52:47 UTC - in response to Message 17524.

Yes, it is a known since Sep 2001
http://primes.utm.edu/top20/page.php?id=5

So now we step into "terra incognita".

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Message 17527 - Posted: 18 Aug 2009 | 17:00:12 UTC - in response to Message 17526.

Yes, it is a known since Sep 2001
http://primes.utm.edu/top20/page.php?id=5

So now we step into "terra incognita".

Primorials were searched up to about 700,000, but I don't know if the entire range was searched and I'm fairly certain that there were no double-checks done. Based upon the resources I've seen thrown at this, it is unlikely that PrimeGrid will get to 700,000 in 2009.

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Message 17528 - Posted: 18 Aug 2009 | 18:00:39 UTC - in response to Message 17527.

Yes, it is a known since Sep 2001
http://primes.utm.edu/top20/page.php?id=5

So now we step into "terra incognita".

Primorials were searched up to about 700,000, but I don't know if the entire range was searched and I'm fairly certain that there were no double-checks done. Based upon the resources I've seen thrown at this, it is unlikely that PrimeGrid will get to 700,000 in 2009.

The last recorded completion limits are:

p# + 1: 637,000 (750,000)
p# - 1: 650,000 (730,000)

In parenthesis are possible limits if all ranges were completed and returned.
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Message 17555 - Posted: 19 Aug 2009 | 14:53:18 UTC - in response to Message 17528.

The last recorded completion limits are:

p# + 1: 637,000 (750,000)
p# - 1: 650,000 (730,000)

I wonder what are completion limits on Factorial?

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Message 18154 - Posted: 24 Sep 2009 | 13:35:33 UTC

Port 12008 is out of work?

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Message 18156 - Posted: 24 Sep 2009 | 16:24:15 UTC - in response to Message 18154.

Port 12008 is out of work?

Sorry

I have loaded a new file now.

Lennart

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Message 19118 - Posted: 13 Nov 2009 | 10:49:20 UTC

Is the Primorial 12008 port down?

I keep getting
[2009-11-13 10:18:27 GMT] prpnet.primegrid.com:12008 connect to socket failed
Also the User and Server Stats pages are not accessible.

(using linux 2.4.5 client)

Lennart SM5YMT
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Message 19126 - Posted: 13 Nov 2009 | 16:15:40 UTC - in response to Message 19118.

Is the Primorial 12008 port down?

I keep getting
[2009-11-13 10:18:27 GMT] prpnet.primegrid.com:12008 connect to socket failed
Also the User and Server Stats pages are not accessible.

(using linux 2.4.5 client)

Sorry It was down.

I was updating the server software and missed to restart that one :(

It's up now

Lennart

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Message 25554 - Posted: 12 Aug 2010 | 12:39:05 UTC

Is the change in crediting points in the Primorial Prime Search done on purpose? This changed after the port ran dry. I see that it first went to some ridiculously high credits per number and was fixed to a lower level but not to the original one.

Around 550 points per number was credited on July 25th, 2010
and now (August 21st, 2010) it is around 32k points per number.

Not that this has any real significance, but the statistics have become a bit funny: they show me leading (because I currently test more numbers than the others (unconnected and SysadmAtNbg)).

On a side note: If I estimated correctly, only about 1000 tests remain on the plus-side before we enter the unknown!
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1784180997819127957596374417642156545110881094717*2^16+1 divides F_14

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Message 25556 - Posted: 12 Aug 2010 | 17:02:38 UTC

I have adressed this here ...
I will look forward what will happen

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Message 25698 - Posted: 20 Aug 2010 | 6:58:52 UTC

I see that the crediting was corrected and credits recalculated. Thank you for that. Now I went back to my rightful place as 4th on the list. :)

I moved the rest of my spare cores from SR5 to PRS for the next week (after that I will not be able to use them anymore). It won't be enough to push the leading edge to the unknown, but at least we get a bit closer to it.
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1784180997819127957596374417642156545110881094717*2^16+1 divides F_14

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Message 26053 - Posted: 1 Sep 2010 | 17:02:17 UTC

Update

A nice milestone was reached yesterday. The Primorial Prime Search reached n=650K. Now both +1 and -1 forms are at first pass level. Congratulations to everyone involved with the project.

For current status, please see PRS stats.
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Message 26116 - Posted: 5 Sep 2010 | 16:16:38 UTC - in response to Message 26053.

I see that in the past four days that we are now nearing 675,000. We should be able to reach about 700,000 by next weekend.

Although the first post states that the coordinated search ended at about 650,000, I do know that some testing was done above that, possibly up to 700,000. I don't know where the next primorial will show up, but it would be cool for the project to find one by the end of the year. I think it would be quite a notable achievement since the last one was found almost a decade ago.

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Message 26118 - Posted: 5 Sep 2010 | 18:33:41 UTC - in response to Message 26116.

Although the first post states that the coordinated search ended at about 650,000, I do know that some testing was done above that, possibly up to 700,000.

This is true. The last known update to the previous search was Feb 8, 2006. At that point the following ranges were listed as complete:
n<637K for p# + 1
n<650K for p# - 1

However, there were open reservations up to the following:

n<750K for p# + 1
n<730K for p# - 1

It is unknown how much of these open reservations were completed. I'll update the first post to include this information as well.
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Message 26377 - Posted: 14 Sep 2010 | 11:27:22 UTC

It looks like the http://prpnet.primegrid.com:12008/server_stats.html page shows wrong Max N values.

Keep up the good work

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Message 26383 - Posted: 14 Sep 2010 | 16:02:03 UTC - in response to Message 26377.

It looks like the http://prpnet.primegrid.com:12008/server_stats.html page shows wrong Max N values.

This is related to the server issues we had earlier. Therefore, we are going to let the server run dry and load new work from scratch. We apologize for the inconvenience. Please have a back-up port available while this one remains empty.
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Message 26841 - Posted: 6 Oct 2010 | 16:21:54 UTC - in response to Message 26383.

Now that a factorial prime has been found, it is time to find a primorial. It has been more than 9 years since the last primorial was found and a -1 primorial hasn't been found in 18 years, which is probably before some of you were born.

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Message 28242 - Posted: 23 Nov 2010 | 15:02:12 UTC - in response to Message 26841.

If the heuristic density of primorial primes holds, we should expect to find (e^gamma)*log(N) primes of each of the forms p#+/-1 with p<N, where gamma is the Euler-Mascheroni constant.

If we extend our search from p=700k to 1M we should therefore expect around 1.27 primorials primes of both forms in that interval.

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Message 28247 - Posted: 23 Nov 2010 | 16:54:49 UTC - in response to Message 28242.

If the heuristic density of primorial primes holds, we should expect to find (e^gamma)*log(N) primes of each of the forms p#+/-1 with p<N, where gamma is the Euler-Mascheroni constant.

If we extend our search from p=700k to 1M we should therefore expect around 1.27 primorials primes of both forms in that interval.

The project should reach p=800k in the next day or two. No new primes to report.

Maybe I should stop putting my resources against it. I did that on GCW13 and it kicked out a base 13 prime a couple days later. I also did it on factorial (switching to all primorial) and the new factorial was a couple of days later. If I quit primorial, that should guarantee a new prime within a couple of days. :-)

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Message 28269 - Posted: 24 Nov 2010 | 8:50:24 UTC - in response to Message 28247.

If my calculations are correct, the heuristic chance of finding a primorial prime in the range 700k-1M exceeds 50% at p=850147 so you might want to wait a few days more before leaving...

Based on the same calculations, the chance of finding a prime in the whole range is ~72%.

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Message 29674 - Posted: 20 Dec 2010 | 8:28:14 UTC

Wow, it looks like I found a PRS.

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Message 29677 - Posted: 20 Dec 2010 | 9:16:51 UTC - in response to Message 29674.

Excellent! Congrats!
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Message 29678 - Posted: 20 Dec 2010 | 9:55:46 UTC - in response to Message 29674.

Congrats!

That's the first primorial prime found in more than 9 years.
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Message 29679 - Posted: 20 Dec 2010 | 13:03:27 UTC - in response to Message 29674.

Wow, it looks like I found a PRS.
I am running a DC on it now but I think it is ok.

Congrats :)

Lennart

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Message 29683 - Posted: 20 Dec 2010 | 14:28:32 UTC

Wow, congrats.
I run primorial 3 days ago...perhaps I missed it just by inch.
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Message 29692 - Posted: 20 Dec 2010 | 18:08:31 UTC

Congratulations!

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Message 29693 - Posted: 20 Dec 2010 | 19:06:05 UTC

Having put some resources to this search earlier, I must say this is truly a nice Christmas present to us!

Congratulations!

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Message 29947 - Posted: 24 Dec 2010 | 15:34:50 UTC

On 20 Dec 2010, 08:05:22 UTC, PrimeGridâ€™s PRPNet found the largest known Primorial prime: 843301#-1

The prime is 365,851 digits long and enters Chris Caldwell's The Largest Known Primes Database ranked 1st for Primorial primes and 260th overall.

The discovery was made by MichaÅ‚ Gasewicz of Poland using an Intel dual Xeon E5520 @ 2.27GHz with 12GB RAM, running 64 bit Linux. This computer took about 71 hours and 20 minutes to complete the primality test using pfgw x64. MichaÅ‚ is a member of the BOINC@Poland team.

For more details, please see the official announcement.
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Message 38283 - Posted: 14 Jul 2011 | 13:30:23 UTC

About 100 tests remain to n=1000000. Nice milestone for project!

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Message 38348 - Posted: 15 Jul 2011 | 18:46:12 UTC

port is out of work

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Message 38362 - Posted: 15 Jul 2011 | 21:17:30 UTC - in response to Message 38348.

port is out of work

Lennart

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Message 50708 - Posted: 28 Feb 2012 | 2:24:09 UTC

Port 12008 will be dry in less than 1 day.

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Message 55248 - Posted: 5 Jun 2012 | 10:52:19 UTC

port is out of new work and challenge is ongoing
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Message 89520 - Posted: 3 Nov 2015 | 12:39:36 UTC

This thread hasn't been updated in a while. Are there any untested gaps below p=2000000 per the stats page: http://prpnet.primegrid.com:12008? I just want to double-check that all p were loaded. It also appears that results for p < 700000 were removed from the server. Is that correct?

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Message 89522 - Posted: 3 Nov 2015 | 14:17:54 UTC - in response to Message 89520.

This thread hasn't been updated in a while. Are there any untested gaps below p=2000000 per the stats page: http://prpnet.primegrid.com:12008? I just want to double-check that all p were loaded. It also appears that results for p < 700000 were removed from the server. Is that correct?

Looking through the completed tests log, I don't see any apparent gaps. Tests for p below 700K were finished by 2010-08-08 and were apparently removed from the server. My oldest backup from 2013-06-22 doesn't have them. That happened before my time here and I don't know if they were removed because of a lack of space, a disk crash or some other reason. But I do have the completed tests log from the very beginning (first listed candidate: 41#-1). Scrolling through the log and examining the database records, there's only one record that looks "iffy". I'm retesting that candidate right now - the existing residue reads as DONE.

The reason there were gaps in the GFN testing was that various versions of genefer were used far past their b-limits. 99.9% of clients did not fall back to a different genefer version or even PFGW, but returned a blank residue which the server accepted. The PRPNet server never sent those candidates out again.

There were never any such issues with PFGW and so the data looks pretty good at first glance.

Edit: That candidate whose residue was DONE is composite.

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Message 89556 - Posted: 4 Nov 2015 | 19:01:23 UTC

No Pending WU's

Looks like everybody is doing boinc. ((((((
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