## Re: LLRP4 Version 3.3 now available !

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• Jean, Thank you for the new version in almost no time! I tried to test Kynea numbers. It s working but it s slower than pfgw. Two examples are given below. Now
Message 1 of 6 , Dec 1, 2004
Jean,
Thank you for the new version in almost no time!

I tried to test Kynea numbers. It's working but it's slower
than pfgw. Two examples are given below. Now I'm testing k*2^n+1.

(2^110614+1)*2^110616-1 = (2^110615+1)^2 - 2 is prime! Time :
929.038 sec. [LLR 3.3, 2.4 GHz P4]

(2^110615+1)^2-2 is 3-PRP! (767.4431s+0.0228s) [pfgw, 2.2 GHz P4]

--------------------
(2^240067+1)*2^240069-1 = (2^240068+1)^2 - 2 is not prime. Res64:
A06A1D3A94955672 Time : 4842.764 sec. [LLR-3.3, 2.4GHz P4]

(2^240065+1)^2-2 is composite: [1FA6F8AA16DC1765] (3412.6447s+0.0320s)
[pfgw, 2.4GHz P4]

Regards,
Predrag
• ... The way to speed up searching Carol/Kynea numbers is through modular reduction. When reducing over 2^n+-2^k-1 ( k
Message 2 of 6 , Dec 1, 2004
--- In primenumbers@yahoogroups.com, "pminovic" <pminovic@y...> wrote:
>
> Jean,
> Thank you for the new version in almost no time!
>
> I tried to test Kynea numbers. It's working but it's slower
> than pfgw. Two examples are given below. Now I'm testing k*2^n+1.
>
> (2^110614+1)*2^110616-1 = (2^110615+1)^2 - 2 is prime! Time :
> 929.038 sec. [LLR 3.3, 2.4 GHz P4]
>
> (2^110615+1)^2-2 is 3-PRP! (767.4431s+0.0228s) [pfgw, 2.2 GHz P4]
>
> --------------------
> (2^240067+1)*2^240069-1 = (2^240068+1)^2 - 2 is not prime. Res64:
> A06A1D3A94955672 Time : 4842.764 sec. [LLR-3.3, 2.4GHz P4]
>
> (2^240065+1)^2-2 is composite: [1FA6F8AA16DC1765] (3412.6447s+0.0320s)
> [pfgw, 2.4GHz P4]
>
The way to speed up searching Carol/Kynea numbers is through modular
reduction. When reducing over 2^n+-2^k-1 ( k<70% of n ingeneral ) we
can use additions and shifts only; There is no need for general
modular reduction. Maybe we need a new library from George Woltman...

Paul
• Thank you for your tests ! I am not surprised if LLPP4 deterministic test is slower than pfgw PRP one, because the Computing U0 loop is more time consuming
Message 3 of 6 , Dec 1, 2004
Thank you for your tests !

I am not surprised if LLPP4 deterministic test is slower than pfgw PRP
one, because the "Computing U0" loop is more time consuming than the
LL loop... Is the deterministic pfgw test also faster ?

Second question : with composite candidates, you found different
residues with pfgw and with LLRP4, it may be normal, or it may be
still an LLR bug...

Regards,

Jean

--- In primenumbers@yahoogroups.com, "pminovic" <pminovic@y...> wrote:
>
> Jean,
> Thank you for the new version in almost no time!
>
> I tried to test Kynea numbers. It's working but it's slower
> than pfgw. Two examples are given below. Now I'm testing k*2^n+1.
>
> (2^110614+1)*2^110616-1 = (2^110615+1)^2 - 2 is prime! Time :
> 929.038 sec. [LLR 3.3, 2.4 GHz P4]
>
> (2^110615+1)^2-2 is 3-PRP! (767.4431s+0.0228s) [pfgw, 2.2 GHz P4]
>
> --------------------
> (2^240067+1)*2^240069-1 = (2^240068+1)^2 - 2 is not prime. Res64:
> A06A1D3A94955672 Time : 4842.764 sec. [LLR-3.3, 2.4GHz P4]
>
> (2^240065+1)^2-2 is composite: [1FA6F8AA16DC1765] (3412.6447s+0.0320s)
> [pfgw, 2.4GHz P4]
>
> Regards,
> Predrag
• ... PRP ... This is true, it takes about 50 minutes to compute U0, I ll append the lresults.txt file tomorrow. ... No! To prove primality of a PRP using pfgw
Message 4 of 6 , Dec 1, 2004
> I am not surprised if LLPP4 deterministic test is slower than pfgw
PRP
> one, because the "Computing U0" loop is more time consuming than the
> LL loop...

This is true, it takes about 50 minutes to compute U0, I'll
append the lresults.txt file tomorrow.

> Is the deterministic pfgw test also faster ?

No! To prove primality of a PRP using "pfgw -tp" is very slow.
Again I don't have exact timings handy but I think at least an
hour in comparison to less than 18 minutes to find that
(2^110615+1)^2-2 is 3-PRP.

> Second question : with composite candidates, you found different
> residues with pfgw and with LLRP4, it may be normal, or it may be
> still an LLR bug...

The input is different too, n=240068 and n=240065. The survival
rate of Kynea (and Carol) is high and there are so many
candidates to test that I simply cannot afford to process the
same number twice :-)) Will try the same number later using
smaller exponents.

BTW, testing k*2^n+1, n~180,000, both the new LLR and PRP3
could process one number in almost exactly the same time, about
66 sec on 2.4GHz P-4.

Regards,
Predrag
• The following is posted on behalf of Jean Penne who sent his reply to primenumbers-owner instead of primenumbers by mistake. Cheers Ken ... pfgw ... the
Message 5 of 6 , Dec 1, 2004
The following is posted on behalf of "Jean Penne" who sent his reply
Cheers
Ken
wrote:
>
> > I am not surprised if LLPP4 deterministic test is slower than
pfgw
> PRP
> > one, because the "Computing U0" loop is more time consuming than
the
> > LL loop...
>
> This is true, it takes about 50 minutes to compute U0, I'll
> append the lresults.txt file tomorrow.
>

> > Is the deterministic pfgw test also faster ?
>
> No! To prove primality of a PRP using "pfgw -tp" is very slow.
> Again I don't have exact timings handy but I think at least an
> hour in comparison to less than 18 minutes to find that
> (2^110615+1)^2-2 is 3-PRP.
>

I am also not surprised : Deterministic pfgw pays for its more
general algorithms than those of LLR.

> > Second question : with composite candidates, you found different
> > residues with pfgw and with LLRP4, it may be normal, or it may be
> > still an LLR bug...
>
> The input is different too, n=240068 and n=240065. The survival
> rate of Kynea (and Carol) is high and there are so many
> candidates to test that I simply cannot afford to process the
> same number twice :-)) Will try the same number later using
> smaller exponents.
>

My fault ! I did'nt see the inputs were different...

> BTW, testing k*2^n+1, n~180,000, both the new LLR and PRP3
> could process one number in almost exactly the same time, about
> 66 sec on 2.4GHz P-4.
>

Again, I am not surprised, PRP3 and LLRP4 use exactly the same code
to do squarings, the only difference is that, for Proth deterministic
tests, LLR computes the base "a" for each number, although PRP3 sets
always "a" = 3, but all that is done outside the loops.

Regards,
Jean
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