A few questions about CHG
 

Hey guys,


So I have a generalized Lucas PRP of ~13k digits, and N-1 has been factored to about 25.55%.

The first question is would this percentage allow the CHG test to be done in a reasonable time?

I would also like to ask, is it feasible to include the factors of N+1 (about 20 total digits) in the process? I've read that considering factors of both N+1 and N-1 requires two CHG processes for factors congruent to 1 and N respectively, so it might be counter-productive. Please correct me if I'm wrong here.

For the third question, I'm thinking about doing the interval selection manually instead of relying on the script. Am I correct in the following phrasing of the whole process?
[LIST][*]Choose several tuples of parameters [TEX][h_i,u_i,\beta_i,\gamma_i][/TEX], such that [TEX]2(\alpha+\beta_i)u_ih_i>u_i(u_i+1)+\gamma_ih_i(h_i-1)[/TEX], and the union of all intervals [TEX][\beta_i,\gamma_i][/TEX] covers [TEX][0,1-3\alpha][/TEX]. ([TEX]\alpha[/TEX] is the factorization percentage of N-1)[*]For each tuple of parameters, construct a matrix as in Corollary 3.1 of the CHG paper, use LLL reduction to find a short vector in its row space, and then attempt to show the polynomial induced by this short vector has no integer roots.[*]The total running time is proportional to [TEX]\sum_{i}h_i^4[/TEX], so this is what I should be minimizing when choosing the intervals.[/LIST]Many thanks for your help!

Is [URL="https://primes.utm.edu/primes/page.php?id=125746"]this[/URL] the number in question? I guess not because it is much bigger than ~13k digits.

You should email Prof. Chis Caldwell to change it to a CHG prover code from a primeform one. Also the syntax is wrong. It should use U(?,-1,?). And the comment should say "Generalized Lucas number".

Congrats. I am interested in the proof method. Please post its output here. :smile:

No, the PRP I was referring to is Phi(3121,-15439) at 13069 digits. Phi(4201,-5798) is a much lighter CHG proof with 28% factored, thanks to a P1740 factor of N-1.

[QUOTE=wpolly;499278]No, the PRP I was referring to is Phi(3121,-15439) at 13069 digits. Phi(4201,-5798) is a much lighter CHG proof with 28% factored, thanks to a P1740 factor of N-1.[/QUOTE]
It is not clear why that would be a Generalized Lucas.

It is, in a sense, a generalized 'Generalized repunit' (but UTM only takes positive bases b>2).Only a few forms (like Wagstaffs) also [I]happen [/I]to be Generalized Lucas, but not just any (b^n+1)/(b+1).

[QUOTE=Batalov;499316]It is not clear why that would be a Generalized Lucas.

It is, in a sense, a generalized 'Generalized repunit' (but UTM only takes positive bases b>2).Only a few forms (like Wagstaffs) also [I]happen [/I]to be Generalized Lucas, but not just any (b^n+1)/(b+1).[/QUOTE]


Actually, (b^n+1)/(b+1) is just U(b-1,-b,n).

Yeah, that's true, but a and b are not irrational and the number looks sort of plain.



Not clear, then, why in the UTM stamp collection Gaussian-Mersennes are not double listed as Generalized unique - but they are.

[QUOTE=Batalov;499320]Yeah, that's true, but a and b are not irrational and the number looks sort of plain.
[/QUOTE]


Indeed. I think these negative-base GRUs should have a separate category, maybe "Generalized Wagstaff"?

The [URL="https://primes.utm.edu/primes/page.php?id=125746"]prime[/URL] is in limbo. Please discuss with Prof. Chris Caldwell what's to be done.

You can also rewrite it in U() form (like [URL="https://primes.utm.edu/primes/page.php?id=125756"]this one[/URL]).
But then again, maybe CC is not working too hard on weekends. Wait for Monday.

[QUOTE=wpolly;498911]Hey guys,

So I have a generalized Lucas PRP of ~13k digits, and N-1 has been factored to about 25.55%.

The first question is would this percentage allow the CHG test to be done in a reasonable time?[/QUOTE]
So, just yesterday I started CHG with 26.85% ratio, and then found a 41-digit ECM factor that allowed to start another CHG with 27.05% ratio. It turns out that the speed is at least 4x faster with 27.05% (at 20k digit size). D.Broadhurst keeps the hardest CHG list somewhere. I believe the record is somewhere around 26%? With lower ratio, thousands iterations will be needed - with enormous time.

Anyway a 13069 number will now be out of top20 and will not be accepted.

wpolly's [URL="https://primes.utm.edu/primes/page.php?id=125746"]number[/URL] has been adjusted and has been accepted as a top20 Gen. Lucas number. However Serge's [URL="https://primes.utm.edu/primes/page.php?id=125756"]larger number[/URL] does not have the right colour for the top20. I think CC has another adjustment to do.