• Hi, Great news! I decided to test Jim s new GMP-ECM and found the following two factors of C233 within a couple hours of each other: Input number is 331...349
Message 1 of 16 , Sep 30, 2001
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Hi,

Great news! I decided to test Jim's new GMP-ECM and found the
following two factors of C233 within a couple hours of each other:

Input number is 331...349 (233 digits)
Using B1=4999999, B2=1447295850, polynomial Dickson(30),
sigma=823490261, Loop 4
Step 1 took 640919ms for 65212548 muls, 3 gcdexts
Step 2 took 290476ms for 27877057 muls, 75678 gcdexts
********** Factor found in step 2:
75518863997370018078536594994311911106041
Found probable prime factor of 41 digits:
75518863997370018078536594994311911106041

Input number is 331...349 (233 digits)
Using B1=4999999, B2=1447295850, polynomial Dickson(30),
sigma=1685681404, Loop 7
Step 1 took 633918ms for 65212548 muls, 3 gcdexts
Step 2 took 268822ms for 27877057 muls, 75678 gcdexts
********** Factor found in step 2: 855877717157262831328185551318041
Found probable prime factor of 33 digits:
855877717157262831328185551318041

After removing both these factors, the cofactor is still composite.
VFYPR quickly proved both of these factors prime. Using these with
David's PRP in PFGW:

Primality testing 15793939750*R(13860)/R(10)+1 [N-1,
Brillhart-Lehmer-Selfridge]
Reading factors from helper file 13680fac.txt
Running N-1 test using base 3
Running N-1 test using base 7
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 30.13%
15793939750*R(13860)/R(10)+1 is PRP! (707.770000 seconds)

And David's KP Pari code at
http://groups.yahoo.com/group/primeform/message/2163 quickly completes
the proof. Now lots of fun can be had with R(13860)!

Greg
• ... Oh, you beauty Greg! Big hand to Jim too! ... My, that s a big B1! How many curves did you run? ... Do we have an independent way of verifying a
Message 2 of 16 , Sep 30, 2001
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On Sun, 30 September 2001, gchil0@... wrote:
> Hi,
>
> Great news! I decided to test Jim's new GMP-ECM and found the
> following two factors of C233 within a couple hours of each other:

Oh, you beauty Greg!
Big hand to Jim too!

> Input number is 331...349 (233 digits)
> Using B1=4999999, B2=1447295850, polynomial Dickson(30),

My, that's a big B1!
How many curves did you run?

> And David's KP Pari code at
> http://groups.yahoo.com/group/primeform/message/2163 quickly completes
> the proof. Now lots of fun can be had with R(13860)!

Do we have an independent way of verifying a certificate of this yet, or is it 'correct by inspection'?

Well done again!

Phil

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• Hi, ... Use two computers, I had run 10 curves on all composites at B1=1000000 and, trying my luck at the casino, about 20 curves on the smallest 7 composites
Message 3 of 16 , Sep 30, 2001
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Hi,

> > Input number is 331...349 (233 digits)
> > Using B1=4999999, B2=1447295850, polynomial Dickson(30),
>
> My, that's a big B1!
> How many curves did you run?

Use two computers, I had run 10 curves on all composites at B1=1000000
and, trying my luck at the casino, about 20 curves on the smallest 7
composites at B1=5000000. I was very surprised when, just before
going to bed last night, I checked the results and found I had gotten
lucky not once but twice on the largest of those 7.

> > And David's KP Pari code at
> > http://groups.yahoo.com/group/primeform/message/2163 quickly
completes
> > the proof. Now lots of fun can be had with R(13860)!
>
> Do we have an independent way of verifying a certificate of this
yet, or is it 'correct by inspection'?

When David arrives, I am sure he will re-run the proof with his code.
Has anyone else written an implementation of KP?

Greg
• ... B1=1000000 ... gotten ... 20? *falls off chair* ... code. ... Konyagin may, or maybe Pomerance! :-) It looks quite simple from David s code though.
Message 4 of 16 , Sep 30, 2001
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> > My, that's a big B1!
> > How many curves did you run?
>
> Use two computers, I had run 10 curves on all composites at
B1=1000000
> and, trying my luck at the casino, about 20 curves on the smallest 7
> composites at B1=5000000. I was very surprised when, just before
> going to bed last night, I checked the results and found I had
gotten
> lucky not once but twice on the largest of those 7.

20?
*falls off chair*

> > > And David's KP Pari code at
> > > http://groups.yahoo.com/group/primeform/message/2163 quickly
> completes
> > > the proof. Now lots of fun can be had with R(13860)!
> >
> > Do we have an independent way of verifying a certificate of this
> yet, or is it 'correct by inspection'?
>
> When David arrives, I am sure he will re-run the proof with his
code.
> Has anyone else written an implementation of KP?

Konyagin may, or maybe Pomerance! :-)

It looks quite simple from David's code though.
However, a reimplementation _based on_ that code may not be any better
than simply rerunning it on a second machine, when it comes to
'independence'.
Oh, I've put a copy/paste of David's code into the files area of the
primenumbers group, under the Pari-GP directory. It needs custmising
for the problem in hand, but it tells you that, so shouldn't confuse
anyone.

Phil
• Great stuff Greg: both from the same target! NB : No more ECM work needed on any of the 17 We are now at 30.15% and so ready for KP on constructs using
Message 5 of 16 , Sep 30, 2001
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Great stuff Greg: both from the same target!

NB : No more ECM work needed on any of the 17

We are now at 30.15% and so ready for KP on constructs
using R(13860):

[13860 0.30125 -17]

n: 495 693 770 924 990 1155 1386 1540 1980 1980
c: 164 293 229 159 229 460 340 212 198 187

n: 2772 3465 4620 4620 6930 13860 13860
c: 694 1410 437 440 1400 1423 1416

I think that, when it is proven, the all-odd-digits pali

should be credited to "Primenumbers e-goup"

Excuse me for not doing the KP for a few days;
it happens that two other projects are almost
at fruition; so expect 3 different anouncements, real soon.

Thanks to everyone and especially our host, Phil

In haste

David
• ... Most of the work is done by Pfgw. My Pari code, which Greg simply ran as is outputs 4 integer coeffs of a cubic poly, P. The KP part of the cert is the
Message 6 of 16 , Sep 30, 2001
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> Do we have an independent way of verifying a
> certificate of this yet

Most of the work is done by Pfgw.
My Pari code, which Greg simply ran "as is"
outputs 4 integer coeffs of a cubic poly, P.

The KP part of the cert is the proof that no positive
integer, n, satisifes P(n)=0.

In GRU work, my exact cubic always agreed with Andy Steward's.

When all 3 roots are real, the proof is trivial:
I find 3 non-integer reals such the 3 *exact* integers

P(floor(real))*P(ceil(real))

are negative. [OK?]

A bit of "if" statementing is needed when there is a pair
of complex roots, but Greg did not hit that case.

However, I believe that Satoshi Tomabechi would we
an appropiate validator. Any chance, please, Satoshi?

David
• David Broadhurst wrote. ... Where are factors of N-1? ( N=R(13860) ) Or please send me help file. I would like to test primality. I implmented all primality
Message 7 of 16 , Oct 1, 2001
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> A bit of "if" statementing is needed when there is a pair
> of complex roots, but Greg did not hit that case.
>
> However, I believe that Satoshi Tomabechi would we
> an appropiate validator. Any chance, please, Satoshi?

Where are factors of N-1? ( N=R(13860) )
Or please send me "help" file.
I would like to test primality.

I implmented all primality tests which are described
in Crandall and Pomerance's book. My code verified
several results about GRU prime by Andy Steward.
My code will give a verification by another implementaion.

Phi Carmody wrote.
>It looks quite simple from David's code though.

My code may also look "too" simple:-)

Satoshi Tomabechi
• David Broadhurst wrote. ... Where is helper file? Or please send me helper file. I would like to test primality. But I have only 13.5% factors of N-1 now. My
Message 8 of 16 , Oct 1, 2001
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> However, I believe that Satoshi Tomabechi would we
> an appropiate validator. Any chance, please, Satoshi?

Where is "helper" file?
Or please send me helper file.
I would like to test primality.
But I have only 13.5% factors of N-1 now.
My code may give a verification by another implementaion.

Phi Carmody wrote.
>It looks quite simple from David's code though.

My code may also look "too" simple:-)

Satoshi Tomabechi
• ... I put it in http://groups.yahoo.com/group/primenumbers/files/Factors/HD13860.fac ... Just what we need, thanks. ... I bet! This simple, uncombined, KP is
Message 9 of 16 , Oct 1, 2001
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> Where is "helper" file?
I put it in
> My code may give a verification by another implementaion.
Just what we need, thanks.
> My code may also look "too" simple:-)
I bet! This simple, uncombined, KP is easy, not so?
Have you coded N+1?
Have you coded 10/3*Max(F1,F2)+Min(F1,F2)>1 ?
Best regards
David
Message 10 of 16 , Oct 2, 2001
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Hi,

> Where are factors of N-1? ( N=R(13860) )
> Or please send me "help" file.
> I would like to test primality.

Umm...looks like David did the same at about the same time :-) but his
is missing the 2 new factors so I'll leave mine for now.

David...could you also update hd1k.zip with the 2 new factors and
Base10.zip with all 3 new factors?

Thanks,
Greg
• ... Sorry about that. I had uploaded an old file. Have now remedied that, with your 2 gems at the top:-) ... Sorry, my file management skills are minimal. It s
Message 11 of 16 , Oct 2, 2001
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Greg Childers wrote:
> looks like David did the same at about the same time
> but his is missing the 2 new factors
Have now remedied that, with your 2 gems at the top:-)
> David...could you also update hd1k.zip with the 2 new factors and
> Base10.zip with all 3 new factors?
Sorry, my file management skills are minimal.
It's already a minor miracle that I managed to
get the previous version more or less OK.
Please Greg, will you take over as Base10 DB manager?
David
• ... OK, I ll delete mine. ... Sure. Delete the files you ve uploaded and I ll upload the updated versions. Greg
Message 12 of 16 , Oct 2, 2001
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> Have now remedied that, with your 2 gems at the top:-)

OK, I'll delete mine.

> > David...could you also update hd1k.zip with the 2 new factors and
> > Base10.zip with all 3 new factors?
> Sorry, my file management skills are minimal.
> It's already a minor miracle that I managed to
> get the previous version more or less OK.
> Please Greg, will you take over as Base10 DB manager?

versions.

Greg
• ... Sure thing, Boss! That folder is now totally Broadhurst free. Thanks for taking over! David
Message 13 of 16 , Oct 2, 2001
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Greg Childers commanded:
> Delete the files you've uploaded
> and I'll upload the updated versions.
Sure thing, Boss! That folder is now
Thanks for taking over!
David
• Thank you, Greg and David. My program shows that 15793939750*R(13860)/R(10)+1 is prime, however I have little confidence in this result:-) I hope that the
Message 14 of 16 , Oct 2, 2001
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Thank you, Greg and David.

My program shows that 15793939750*R(13860)/R(10)+1 is prime,
however I have little confidence in this result:-)
I hope that the result is as same as David's.

The coefficients of cubic equation t3X^3 + t2X^2 + t1X + t0 = 0 are

t3 = 9499179946...9373462290 1400 digits
t2 = 2983442091...4520303837 2765 digits
t1 = -5786178299...9234821312 4165 digits
t0 = -8297037200...6079883289 2764 digits

The equation has three real roots r1,r2,r3.

r1=-1.433940810...*10^{-1401}

integral part of r2 246804372...4038770926 ( 1382 digits)
fractional part of r2 0.9565089129....

integral part of r3 -246804372...8946383526 ( 1382 digits)
fractional part of r2 0.5941852755....

Nearest integers of r1,r2 and r3 are not root of the equation.

# It was not easy to compute roots of the cubic equation.

Best regards.

Satoshi Tomabechi
• ... Yes, of course. Furthermore I coded modified KP with additional square tests by you. http://groups.yahoo.com/group/primeform/message/2142 But I have not
Message 15 of 16 , Oct 2, 2001
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> Have you coded N+1?
Yes, of course.
Furthermore I coded modified KP
with additional square tests by you.
http://groups.yahoo.com/group/primeform/message/2142

But I have not coded combined KP.

> Have you coded 10/3*Max(F1,F2)+Min(F1,F2)>1 ?
Sorry, I don't know such condition.
Please tell me reference or pointer to reference.

Satoshi Tomabechi
• ... By F1 and F2 I meant the fractions of N-1 and N+1. When F1 F2 this condition is what I used for Fib(25561), but it also works for F2 F1. Hence the Max and
Message 16 of 16 , Oct 2, 2001
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Satoshi TOMABECHI wrote:
> > Have you coded 10/3*Max(F1,F2)+Min(F1,F2)>1 ?
> Sorry, I don't know such condition.
By F1 and F2 I meant the fractions of N-1 and N+1.
When F1>F2 this condition is what I used for Fib(25561),
but it also works for F2>F1. Hence the Max and Min.
David
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