User group for PFGW & PrimeForm programs
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Re: Gaussian primorial and factorial primes
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Andrew.Walker@ipaustralia.gov.au Message 1 of 7 Dec 20, 2010
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On Dec 19, 2010, at 6:57 AM, mikeoakes2 wrote:
>
> Typing 0,1,2,3,4,5,9,10,11,13,24,65,76 into the OEIS search box gives
> http://oeis.org/A046029
> "Numbers n such that (n!)^2+1 is prime"
> So these have been investigated already, and no more found up to
> n=10000, as there is the comment
> "a(14) > 10000. - Charles R Greathouse IV, Nov 16 2006"
>
> That's rather a poor harvest!
> Anyone up to finding some more?
> (The 2 PFGW runs each took 24 hours on a 1.8 GHz core.)
>
> -Mike Oakes
>
>
>
I looked at a number of these "cyclotomic" forms for the factorial case at
least 4
or 5 years ago now. I have a feeling I left a results file in the file
section of one of these groups,
if I can't find it I'll dig it out at home. Mark, I'm fairly sure you
added this form to one
of your sieve programs for me way back then!
Andrew
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djbroadhurst Message 2 of 7 Dec 20, 2010
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--- In primeform@yahoogroups.com,
Andrew.Walker@... wrote:
> I looked at a number of these "cyclotomic" forms for
> the factorial case at least 4 or 5 years ago now.
http://oeis.org/A083341
gives a broken link to one of your old files:
www.uow.edu.au/~ajw01/ecm/pluspp.txt
David
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mgrogue@wi.rr.com Message 3 of 7 Dec 20, 2010
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> I looked at a number of these "cyclotomic" forms for the factorial case at
> least 4
> or 5 years ago now. I have a feeling I left a results file in the file
> section of one of these groups,
> if I can't find it I'll dig it out at home. Mark, I'm fairly sure you
> added this form to one
> of your sieve programs for me way back then!
That is possible. MultiSieve does support cyclotomics, but not those of factorial or primorial form. MultiSieve would be painfully slow compared to any new code I would write based upon psieve. Geoff Reynolds helped me quite a bit with his x86 wizardry.
It would be fairly easy to extend psieve to support a number of factorial/primorial forms. The source can be found here:
http://pgllr.mine.nu/software/psieve/
fsieve, which is for factorials, can be found here:
http://pgllr.mine.nu/software/fsieve/
It can be build with MinGW on Windows or gcc on other platforms (Mac, Linux) and runs on x86 and ppc. I'm using a "hacked" version of psieve in an effort to disprove Puzzle 10 over at primepuzzles. Steven Harvey is helping me with that.
--Mark
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andrew_j_walker Message 4 of 7 Dec 20, 2010
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--- In primeform@yahoogroups.com, "djbroadhurst" wrote:
>
>
>
> --- In primeform@yahoogroups.com,
> Andrew.Walker@ wrote:
>
> > I looked at a number of these "cyclotomic" forms for
> > the factorial case at least 4 or 5 years ago now.
>
> http://oeis.org/A083341
> gives a broken link to one of your old files:
> www.uow.edu.au/~ajw01/ecm/pluspp.txt
>
> David
>
That link should be removed if anyone can edit it, nearly 5 years
I was at uni! The file I was thinking of is in the file section
of this group, prime-deposit folder, MiscF.txt , last edited 2003 (!)
Andrew
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djbroadhurst Message 5 of 7 Dec 21, 2010
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--- In primeform@yahoogroups.com,
"andrew_j_walker" wrote:
> in the file section of this group, prime-deposit folder, MiscF.txt
Well found, Sir.
http://groups.yahoo.com/group/primeform/files/prime-deposit/MiscF.txt
> phi(5,n!) n!^4+n!^3+n!^2+n!+1
> 1,2,4,5,21,44,64,(244),(268) [1721]
and (268) was the subject of
Exercise 2. Attempt ECM to increase the help for
Calling N-1 BLS with factored part 25.33% and helper 0.18%
(76.21% proof)
Phi(5,268!) is Fermat and Lucas PRP! (2.6870s+0.0036s)
in the hope of a CHG proof.
but unfortunately that seems to be too hard. So please
replace "268" by "244":
Exercise 2. Use CHG to prove the primality of
Calling N-1 BLS with factored part 25.70% and helper 0.58%
(77.73% proof)
Phi(5,244!) is Fermat and Lucas PRP! (2.2092s+0.0020s)
which is less demanding.
David (failed puzzle setter)
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mikeoakes2 Message 6 of 7 Dec 21, 2010
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--- In primeform@yahoogroups.com, "andrew_j_walker" wrote:
>
> --- In primeform@yahoogroups.com, "djbroadhurst" wrote:
> >
> > --- In primeform@yahoogroups.com,
> > Andrew.Walker@ wrote:
> >
> > > I looked at a number of these "cyclotomic" forms for
> > > the factorial case at least 4 or 5 years ago now.
> >
> > http://oeis.org/A083341
> > gives a broken link to one of your old files:
> > www.uow.edu.au/~ajw01/ecm/pluspp.txt
> >
> > David
> >
>
> That link should be removed if anyone can edit it, nearly 5 years
> I was at uni! The file I was thinking of is in the file section
> of this group, prime-deposit folder, MiscF.txt , last edited 2003 (!)
>
Andrew,
Your searches as recorded in
http://groups.yahoo.com/group/primeform/files/prime-deposit/MiscF.txt
had yielded the following:
Long form Prime n values [limit searched]
(n!)^2+n!+1 1,2,3,4,6,76,2837,6001,7076 [7964]
(n!)^2+1 1,2,3,4,5,9,10,11,13,24,65,76 [10632]
(n!)^2-n!+1 2,3,5,7,38,2319,2996,3321,3892 [5500]
These extend a bit the data points I posted in the last couple of days:
Long form Prime n values [limit searched]
(n!)^2+n!+1 0,1,2,3,4,6,76,2837 [4400]
(n!)^2+1 0,1,2,3,4,5,9,10,11,13,24,65,76 [5550]
(n!)^2-n!+1 2,3,5,7,38,2319,2996,3321,3892 [4400]
I'm glad there's no discrepancy!
Mike
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djbroadhurst Message 7 of 7 Dec 21, 2010
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--- In primeform@yahoogroups.com,
"djbroadhurst" wrote:
http://groups.yahoo.com/group/primeform/files/prime-deposit/MiscF.txt
> phi(5,n!) n!^4+n!^3+n!^2+n!+1
> 1,2,4,5,21,44,64,(244),(268) [1721]
To these, we may add (2415), at 28493 digits:
Running N-1 test using base 2417
Running N-1 test using base 2441
Running N+1 test using discriminant 2473, base 72+sqrt(2473)
Calling N-1 BLS with factored part 25.07% and helper 0.03%
(75.24% proof)
Phi(5,2415!) is Fermat and Lucas PRP! (677.9536s+0.0069s)
with poor prospects of a CHG proof.
David
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