R252 is proven

[URL="http://www.mersenneforum.org/posthistory.php?p=208595"][I][COLOR=seagreen]Last fiddled with by Batalov[/COLOR][/I][/URL][I][COLOR=seagreen]; 16 Mar 10 at 10:32 PM Reason: I thought this thread was "up to 256"; please move to 251+[/COLOR][/I]
[I][COLOR=#2e8b57][B]mdettweiler: [/B]moved[/COLOR][/I]

R441, S441, R472 are proven
 

R441, S441, R472 are proven.
To save space, all three in one zip.

S405 is reserved. One k=106 remains.
Reserving R450 as new. One k=57 remains.
Data is attached.

Nice work everyone in figuring out those k's with algebraic factors to make a full covering set on various bases.

It's definitely a big "hole" in the pages right now that I'll be looking at over the next 1-2 days.

That's interesting that the "new" kind of algebraic factors are always k*b = a perfect square.

Willem, your analogy on R40 looks correct. I only checked the k's that you found that could be eliminated and no others.

Edit: I've now checked all k's on the base. Nothing else found.

R361
 

Reserving R361 with conjectured k=8870 as new to 25K.
At n=1500, there are only 45 [I]k[/I]'s left. 44... 43...
Initial files are attached.
[SIZE=1](Two R19 primes are included; they were the smallest for R19, so they are the smallest for R361 as well.)[/SIZE]
________

Gary, if you have all R19 primes, could please PM?
Of interest are somewhat larger even [I]n[/I]'s, except for two odd [I]n[/I]'s for k=138 and [strike]366[/strike]. The small primes of course will be soon rediscovered anyway.

[quote=Batalov;208042]b=294=6*7[sup]2[/sup]

k=6: for even n, divisible by 5; for odd n=2m-1,
6*294[sup]2m-1[/sup]-1 = 6*(6*7^2)[sup]2m-1[/sup]-1 = 6[sup]2m[/sup] * 7[sup]2n[/sup] - 1[sup]2[/sup] = Difference of squares.

k=96=6*4[sup]2[/sup]: ditto.

qed :smile:[/quote]

Stop those pencil proofs! lol :smile:

Good work as usual.

[quote=Batalov;208082]
Now I'd like to get back to the earlier argument: should the sieve [I]or[/I] pfgw remove such cases by a fast factorization of [I]k[/I] and [I]b[/I]?
I think, both!
Or [I]the script[/I].
This is because when people start a new base, they initially use pfgw and [I]the script[/I]. They don't even get to the srsieve until much later.[/quote]

IMHO, it should be in the sieving software. After all, sieving software is for finding factors. To "generalize" its use further, it should find both "numeric" and algebraic factors as best as it can. Similarly, primality proof software is for proving primes, not finding factors.

PFGW as a primality proof program really does us a kindness by having a trial factoring option but that is only a "nicety" option that avoids the extra effort of sieving teeny n-ranges. IMHO, it's usefulness would be virtually as good without such an option.

Even if the sieving software only found "very simple" algebraic factors such as where k and b are both a perfect square/cube/5th power/etc., then its usefulness would be increased. From there, you could then expand its usefullness by having it find "somewhat simple" algebraic factors as shown in the "Generalizing algebraic factors for Riesel bases thread" followed by the "medium difficulty" algebraic factors as you guys have recently found for odd n's where k*b is a square, and then finally finding algebraic factors for cubes and higher powers where there is no very simple pattern and where k and/or b are not necessarily perfect powers of any kind.

All of this said, my/our starting bases script should also be able to eliminate k's with partial algebraic factors to make a full covering set. Why? Because we don't sieve when starting new bases. We (at least me anyway) only do trial factoring to 100% using the -f switch. It is something on the backburner in my head and will be a fairly major enhancement. I will likely start the process as shown above by doing the easy ones where both k and b are perfect squares/cubes/high powers first as a new version 5.0. From there, I'll add them as shown above for versions 5.1, 5.2, etc. Likely I won't do the final one because the situations in which they will apply to this project will be quite rare.


Gary

...I have already started groking the sr(x)sieve code to make some insightful (as usual) additions... :rolleyes:

I always use srsieve to "confirm" (empirically, of course) the conjectured [I]k[/I] -- I set max_k to the conjectured[I] k[/I] and it stays in the pl_remain.txt and is then fed to the srsieve, which in turn doesn't even produce an .npg file with the message that all candidates were eliminated. After the patch, it will do that for some other (partial-trivial+algebraic) [I]k[/I]'s as well; it will also make smaller .npg files in the partial-but-not-complemented-by-trivial algebraic cases.
Which, I think, will be the desired behaviour.
___________

[COLOR=green]R361 is down to only 24 [I]k[/I]'s (at n~=15K in terms of base-19) and may shed some more if you will find some R19 primes from the archives? /wink-wink-nudge-nudge/[/COLOR]

[QUOTE=Batalov;208853]...I have already started groking the sr(x)sieve code to make some insightful (as usual) additions... :rolleyes:

I always use srsieve to "confirm" (empirically, of course) the conjectured [I]k[/I] -- I set max_k to the conjectured[I] k[/I] and it stays in the pl_remain.txt and is then fed to the srsieve, which in turn doesn't even produce an .npg file with the message that all candidates were eliminated. After the patch, it will do that for some other (partial-trivial+algebraic) [I]k[/I]'s as well; it will also make smaller .npg files in the partial-but-not-complemented-by-trivial algebraic cases.
Which, I think, will be the desired behaviour.
___________

[COLOR=green]R361 is down to only 24 [I]k[/I]'s (at n~=15K in terms of base-19) and may shed some more if you will find some R19 primes from the archives? /wink-wink-nudge-nudge/[/COLOR][/QUOTE]

I suggest that you give those changes to Geoff Reynolds. I know that he is very busy, but I suspect that he would welcome the input.

Of course.

[quote=Batalov;208603]S405 is reserved. One k=106 remains.
Reserving R450 as new. One k=57 remains.
Data is attached.[/quote]

I need a search limit on these. Also, is R450 still reserved?

One more thing: Can you please slow down on the new bases for a few days? I need to catch my breath. We have so much work to do on bases that have already been started.