mersenneforum.org The factorization of primorials +/- 1
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 2007-08-10, 07:29 #1 Joppe_Bos   Apr 2007 1001002 Posts The factorization of primorials +/- 1 Hi all, As a hobby project I extended the tables for the primorials +/- 1 from the World Integer Factorization Center. Right now I have the tables with n <= 160. I did quite some amount of ECM work on the extensions as well as one the original tables (and managed to find quite some prime factors). I have mailed Hisanori Mishima but not all my new factors for the original list are on the WIFC website yet. If anyone is interested in helping out please report ECM work done or factors found in this thread. I made a very basic website showing how many curves I have done on these numbers together with the tables at primorial.unit82.com. Furthermore I am quite interested in any theory about the factorization of these numbers. When searching for some articles I managed to find only a few about the primorial primes, so if anyone knows some good resources please let me know! Thanks, Joppe
 2007-08-11, 02:14 #2 jasong     "Jason Goatcher" Mar 2005 5·701 Posts I'm reserving 76#+1 here (Edit: and 82#+1), since I figure this is the most obvious place. If someone wants to give me editing privileges, I'll restrict myself to this thread and only handle the tracking.(I would suck as a mod, but I'm sure people already know that) Last fiddled with by jasong on 2007-08-11 at 02:25
 2007-08-11, 02:32 #3 jasong     "Jason Goatcher" Mar 2005 1101101100012 Posts If I may ask a stupid question. How do you put comments in an ecm input file?
 2007-08-11, 02:42 #4 Citrix     Jun 2003 157410 Posts I have always wondered about this, but never had the time to implement this... If we take P#+1 or P#-1 and do a P-1/P+1 test using B1=P, how many factors would be end up finding? What would be the distribution of such factors? Are any such factors known for large P?
 2007-08-11, 04:35 #5 Citrix     Jun 2003 110001001102 Posts The only solution I know of is 2#+1 is divisible by 3. Are there any more?
2007-08-11, 10:15   #6
Joppe_Bos

Apr 2007

2416 Posts

Quote:
 Originally Posted by jasong I'm reserving 76#+1 here (Edit: and 82#+1), since I figure this is the most obvious place. If someone wants to give me editing privileges, I'll restrict myself to this thread and only handle the tracking.(I would suck as a mod, but I'm sure people already know that)
Thanks for the help jasong! You probably want to reserve E_76 = 383# + 1 and E_82 = 421# + 1. Just report the amount of ECM curves here so I can update the stats on the website. My first goal is to ECM all the numbers up to 40 digits and then 45 (and at the same time focus on the smaller n in #p_n +/- 1 (where p_n is the nth prime).

Quote:
 Originally Posted by jasong If I may ask a stupid question. How do you put comments in an ecm input file?
Lines beginning with a "#" are comments as far as I know.

Last fiddled with by Joppe_Bos on 2007-08-11 at 10:17 Reason: Typo

2007-08-11, 11:14   #7
Joppe_Bos

Apr 2007

2416 Posts

Quote:
 Originally Posted by Citrix I have always wondered about this, but never had the time to implement this... If we take P#+1 or P#-1 and do a P-1/P+1 test using B1=P, how many factors would be end up finding? What would be the distribution of such factors? Are any such factors known for large P?
Quote:
 Originally Posted by Citrix The only solution I know of is 2#+1 is divisible by 3. Are there any more?
Maybe I don't understand your question right but aren't all primorials +/- 1 a solution? For example take a number p_n# + 1 (the nth prime) than factoring with Pollard p-1 algorithm with a bound p_n will always give the trivial factor p_n# + 1 itself (the same holds for p_n# - 1 and using Williams p + 1 algorithm with bound p_n).

About the non-trivial factors I have no idea about the distribution (interesting question!), the first non-trivial factor which can be found with Pollard p-1 with pound p_n is when n = 7 so p_7# + 1 = 17# + 1 = 510511 which has a factor 2 * 3^2 + 1.

 2007-08-11, 12:15 #8 rogue     "Mark" Apr 2003 Between here and the 22·33·5·11 Posts Set up an ECMNet server and you will probably have many more helpers.
2007-08-11, 18:55   #9
Citrix

Jun 2003

2·787 Posts

Quote:
 Originally Posted by Joppe_Bos Maybe I don't understand your question right but aren't all primorials +/- 1 a solution? For example take a number p_n# + 1 (the nth prime) than factoring with Pollard p-1 algorithm with a bound p_n will always give the trivial factor p_n# + 1 itself (the same holds for p_n# - 1 and using Williams p + 1 algorithm with bound p_n).
only if P#+1 is prime. Hence I was asking for the distribution of these factors.

In short find n such that P#+1== 0 (mod n) and the largest prime divisor of n-1 is smaller or equal to P. I tested all n up to 100K, but did not not find a solution except n=3.

Similarly, find x such that P#-1==0 (mod x) and the largest prime divisor of x+1 is smaller or equal to P. No solutions for this.

2007-08-11, 20:18   #10
Citrix

Jun 2003

2·787 Posts

Quote:
 Originally Posted by Citrix In short find n such that P#+1== 0 (mod n) and the largest prime divisor of n-1 is smaller or equal to P. I tested all n up to 100K, but did not not find a solution except n=3. Similarly, find x such that P#-1==0 (mod x) and the largest prime divisor of x+1 is smaller or equal to P. No solutions for this.
Never mind this, a bug in my code, found several solutions for this.

2007-08-11, 23:17   #11
jasong

"Jason Goatcher"
Mar 2005

350510 Posts

Quote:
 Originally Posted by rogue Set up an ECMNet server and you will probably have many more helpers.
Um, because of my impulsiveness and the difficulty of compiling ecm on my quad-core, I'd like to unreserve my numbers.

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