- --- "David Broadhurst <d.broadhurst@...>" <d.broadhurst@...> wrote:
> > Woh! I can't see Dickman in that at all.

...

> > What am I missing?

>

> Let me backtrack a bit, to Aurifeuille.

> (more or less) T1 and T2 digits then Dickman suggests

Spot on. I like it. That's a completely sideways way of looking at things!

> (don't eek me, I only said sugests) that we will

> get them twice as fast, compared with when there

> isn't such a factorization.

I like it!

> Now what's special about those 3 buttons?

Speak for yourself. I used {1,2,12} and {1,4,6} /precisely/ for that reason.

>

> ? print(factor((1+x)*(1+2*x)*(1+12*x)-1))

> [x, 1; 4*x + 3, 1; 6*x + 5, 1]

>

> ? print(factor((1+x)*(1+2*x)*(1+15*x)-1))

> [x, 1; 3*x + 2, 1; 10*x + 9, 1]

>

> ? print(factor((1+x)*(1+4*x)*(1+6*x)-1))

> [x, 1; 2*x + 1, 1; 12*x + 11, 1]

>

> Of course you and I did not bother about this when we used CRT.

I think one of my results was a {1,4,6} wasn't it?

> But Markus is stuck with his NewPgen output and so

Indeed it is.

> can't use CRT. So I thought it would be useful

> to give {1,2,12} a proof in principle by beating the

> 4-Carmichael 4th factor record.

>

> In fact I did it twice over:

>

> 2763260532*((1591638066432*3061#+19)^2-1)*3061#/

> 286610757607008951353515277095+1 3909 p44 03

> 4-Carmichael factor (4)

>

> 757849549*((72753556704*3061#+19)^2-1)*3061#/

> 56731664597567983567442428202862519351615138+1 3891 p44 03

> 4-Carmichael factor (4)

>

> finding ECM factors at about twice

> the rate for this seed as compared with

> the other seeds in my plantpot.

>

> OK, it's a trivial [and suspect] use of Dickman.

It's a great idea for trying to squeeze out factors quicker, certainly.

I'm surprised it works, because the factors need to be of a particular

residue in order to make the thing carmichael (like your lucky 2s in the

3-carmichael).

There's the Saarinen DLOG trick one can use. However, he's not published

that yet. (It's one of the tricks that helped him generate the multi-

billion-digit Carmichael so easily last month.)

You seem to have claimed a *2 speedup over the naive technique? I think I

claimed either a *4 or *6 speedup. However, as you have noted elsewhere I

become constipated with CRT. (Not every element in my speedup contributed a

linear factorisation, and therefore caused doping-bloat.)

Phil

=====

Is that free as in Willy or as in bird?

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http://mailplus.yahoo.com - Phil:

> That's a completely sideways way of looking at things!

Thank ye, kind sir.

> I like it!

> I think one of my results was a {1,4,6} wasn't it?

Yep. I said you didn't need to bother, not that you

didn't bother.

> I'm surprised it works, because the factors need to

\begin{explanation}

> be of a particular residue in order to make the thing

> carmichael

lcm{1,2,12}=3*4

We get the 3 from 4*x+3, since

x=0 mod 3, because of the primorial.

We get the 4 from (1+1/d).

All possible d's are odd.

So half of the combinations of ECM primes work.

But we double these using the 5 from 6*x+5,

since x=0 mod 5, because of the primorial.

So {1,2,12} is as rich as {1,2,3}.

Except that it's twice as fast

as {1,2,3} for ECMing.

\end{explanation} - Given the 1,2,15 seed, or 1,2,12 seed etc

Couldn't we check for a 4th element using ECM, and a 5th element (e) such

that we can form a 5-carm of the form N = e*1*2*15*ECMFactor

Carms of that form

5,7,13,193,439,9637697

5,7,13,193,499,2657,42829

5,19,37,547,1009,3211937

7,19,37,541,601,58741

7,79,157,2341,3541,9181

13,19,37,541,631,2689

13,19,37,541,739,811,1231

13,19,37,541,1009,2311

17,37,73,1093,2081,65521

17,37,73,1093,93809

With my dataset i will have 4 billion possible e values and 350 titanic

seeds of whatever 3 part seed that is picked. Is it possible to create a

carm by extending a "seed" at both ends. IE creating a custom formula for

each e*1*2*15 which is then ecm'd. to get a prime that

makes e*1*2*15*ECMFactor PRP

Markus

>But Markus is stuck with his NewPgen output and so

>can't use CRT. So I thought it would be useful

>to give {1,2,12} a proof in principle by beating the

>4-Carmichael 4th factor record.

>

>In fact I did it twice over:

>

>2763260532*((1591638066432*3061#+19)^2-1)*3061#/

>286610757607008951353515277095+1 3909 p44 03

>4-Carmichael factor (4)

>

>757849549*((72753556704*3061#+19)^2-1)*3061#/

>56731664597567983567442428202862519351615138+1 3891 p44 03

>4-Carmichael factor (4)

>

>finding ECM factors at about twice

>the rate for this seed as compared with

>the other seeds in my plantpot.

>

>OK, it's a trivial [and suspect] use of Dickman.

>

>But it was funny that you should mention him

>just after I had commented on another such seed

>{1,2,15}. But Markus can't use that

>Dickman-enhanced seed, since he

>can't get x=1 mod 5 in

>

>? print(factor((1+x)*(1+2*x)*(1+15*x)-1))

>[x, 1; 3*x + 2, 1; 10*x + 9, 1]

>

>because he used primorial mode in NewPgen.

>

>David

>

>

>Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

>The Prime Pages : http://www.primepages.org/

>

>

>

>Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ - Phil Carmody wrote:
>

Phil,

> ... brilliant numbers. If someone has kept a full record of all the numbers

> that were in the range they tested for brilliantness, then that would be a

> fair corpus, surely? I'm assuming they weren't sieved terribly deeply, and

> therefore only the medium sized factors would need to have been recorded, as

> the small ones can be reproduced easily (hey, I'll offer to find all the

> small factors to test my new factoring algorithm!). Sure they're smaller

> than 200 digits by a factor of 2, but they're about as unbiased as you can

> get, and there's a fair number of them.

>

> Anyone with such a collection?

>

> Phil

>

> =====

> Is that free as in Willy or as in bird?

I have been keeping such a list and was going to offer up all my

findings once I have found my next brilliant number. The program I used

to do the "sieving" was Miracl's factor.exe (slightly modified by me)

and it does some ecm to remove up to 25 digit factors. So, would my

list be too sparse to be useful? I didn't keep any of the sieved

numbers, but everything that fell through that I have nfsx or ppsiqs

factorings of.

I was also thinking of giving the timings for the work thus far, and

maybe also giving a list of the primes that were found in this range.

Would either of the last two pieces of information be useful to anyone?

I'm still compiling the information and still waiting for the next

brilliant, but as soon as both are ready I'll e-mail the list.

-David C. - David Cleaver asked:
> So, would my list be too sparse to be useful?

I think that what you have so far cleaved/cleft/cloven

is rather impressive and that the more you document it

the better. Since the files are presumably not huge,

I can't see that Phil is going to fret if you

create a "brilliant" folder in

http://groups.yahoo.com/group/primenumbers/files/

and add whatever you think might be informative therein.

David