Bases 501-1030 reservations/statuses/primes

[MyDogBuster]

Riesel Base 949
Conjectured k = 56
Covering Set = 5, 97
Trivial Factors k == 1 mod 2(2) and k == 1 mod 3(3) and k == 1 mod 79(79)

Found Primes: 18k's File attached

Trivial Factor Eliminations: 9k's

Conjecture Proven

[MyDogBuster]

Riesel Base 964
Conjectured k = 194
Covering Set = 5, 193
Trivial Factors k == 1 mod 3(3) and k == 1 mod 107(107)

Found Primes: 123k's File attached

Remaining k's: Tested to n=25K
9*964^n-1 <------ Proven composite by partial algebraic factors
129*964^n-1
141*964^n-1
144*964^n-1 <------ Proven composite by partial algebraic factors

Trivial Factor Eliminations: 65k's

Base Released

[MyDogBuster]

Riesel Base 969
Conjectured k = 96
Covering Set = 5, 97
Trivial Factors k == 1 mod 2(2) and k == 1 mod 11(11)

Found Primes: 41k's File attached

Remaining k's:
4*969^n-1 <------ Proven composite by partial algebraic factors
64*969^n-1 <------ Proven composite by partial algebraic factors

Trivial Factor Eliminations: 4k's

Conjecture Proven

[MyDogBuster]

Riesel Base 984
Conjectured k = 196
Covering Set = 5, 197
Trivial Factors k == 1 mod 983(983)

Found Primes: 180k's File attached

Remaining k's: Tested to n=25K
4*984^n-1 <------ Proven composite by partial algebraic factors
9*984^n-1 <------ Proven composite by partial algebraic factors
18*984^n-1
49*984^n-1 <------ Proven composite by partial algebraic factors
64*984^n-1 <------ Proven composite by partial algebraic factors
81*984^n-1
86*984^n-1
99*984^n-1
119*984^n-1
120*984^n-1
121*984^n-1
144*984^n-1 <------ Proven composite by partial algebraic factors
169*984^n-1 <------ Proven composite by partial algebraic factors
191*984^n-1

Base Released

I'll be doing the rest of b=4 mod 5 (larger ck's and stuff I missed) over time. Not reserving anything, for now. Time for a break.
By my count, there are 28 left, most have a ck of >200. I still owe R864.

[KEP]

Originally Posted by henryzz

1/3 needs to be sieved but half of that will quickly be sieved away by the factor 17
the k does need to be tested as far as i can tell although i wouldnt do anything serious until gary posts
to remove the ns with algebraic factors use the perl script in post #64 of this thread to remove ns that are 0 mod 2 and 0 mod 3
i have attached a file sieved to 1e10 for that k with the algebraic factors removed

Thanks for the sieve file, and thanks for your input and help. Your skills is far greater than mine currently. However, I'm wondering now, is there a way to determine weather or not a base contain k's with algebraric factors (such that I can avoid choosing bases in the future of wich there is algebraric factors for some k's)?

Thanks again.

Regards

KEP

[henryzz]

Originally Posted by KEP

Thanks for the sieve file, and thanks for your input and help. Your skills is far greater than mine currently. However, I'm wondering now, is there a way to determine weather or not a base contain k's with algebraric factors (such that I can avoid choosing bases in the future of wich there is algebraric factors for some k's)?

Thanks again.

Regards

KEP

Now thats beyond me
I would actually like to find a k which algebraic factors make as low weight as yours myself so i would like to know what the conditions are.

[gd_barnes]

ALL Riesel bases have squared k's with partial algebraic factors. The base doesn't matter (as long as the conjecture is > 4). Just think of k=4 or 9 on any Riesel base. For k=4, all even n's are eliminated by [2*b^(n/2)-1]*[2*b^(n/2)+1]. The only time algebraic factors can eliminate a k is if the rest of the n's are eliminated by a single factor (frequently 5) or a covering set of factors.

Just let srxsieve do its/their thing. If you get the "x*xx^n-1 contains algebraic factors", it means you can manually remove some (actually very few) n-values. If you have a squared k, you can manually remove the few even n's left over. If you have a cubed k, you can manually remove the fewer n==(0 mod 3) left over. It does NOT mean you can fully eliminate the k from testing.

k=64 happens to be one where you can remove both even n's and n==(0 mod 3) since it's a perfect square and cube. But...it's not just Riesel base 633. You can remove those n-values for k=64 on ALL Riesel bases. On the Sierp side, you could only eliminate n==(0 mod 3) for k=64 because k^2*b^2+1 does not algebraically factor whereas k^2*b^2-1 factors to (k*b-1)*(k*b+1).

Personally, I don't bother with manually removing n-values because most of my testing has been n<=25K. I'd suggest the same for others. It's not worth the personal time to do it for a small amount of CPU time savings. I'd suggest doing the above only for n>25K.

The factorization for the various n-values for 64*633^n-1 is normal. On a fair-sized percentage of k's remaining at CRUS, you'll get down to only 1 n-value out of every 6 or 12 n's remaining where there is not a small factor. That low weight is why those k's are remaining. And the fact that a squared k cannot have an even-n Riesel prime is why more squared k's are remaining than one would expect by chance alone.

In a synopsis: On a squared k, after doing a "mental" elimination of even n's, if you can't quickly determine a covering set of small factors (generally < 10K) for odd n's, then just sieve it normally and test it normally. You'll usually only end up testing about 1-3% more n-values than you would if you manually eliminated the even n's.


Gary

[gd_barnes]

Originally Posted by KEP

Thanks for the sieve file, and thanks for your input and help. Your skills is far greater than mine currently. However, I'm wondering now, is there a way to determine weather or not a base contain k's with algebraric factors (such that I can avoid choosing bases in the future of wich there is algebraric factors for some k's)?

Thanks again.

Regards

KEP

That's what the "generallizing algebraic factors on Riesel bases" thread is for. Think Riesel bases that are:

b==(4 mod 5)
b==(12 mod 13)
b==(16 mod 17)
b==(28 mod 29)
b==(36mod37)
b==(40mod41)
b==(52mod53)
b==(60mod61)
b==(112mod113)

These are all of the base modulos that have been found so far that can possibly contain k's with "partial algebraic factors" that make a full covering set. Base 633 does not fit any of them.

For non-math type people, I put this in as "English" of terms as possible. Please see the smiley face in the first post in that thread here.

Ian has been kind enough to test ~85-95% of the bases where b==(4 mod 5) so that others don't run into the problems that they entail. The others are less common and require that the conjecture be higher than the lowest possible k that can contain partial algebraic factors to make a full covering set. Frequently, it is not so even if a base hits one of the above criteria, it won't have k's to be concerned about.

Generally, don't worry about it. Just check and see if your base fits one of the modulos in that thread. If it does, I'll be glad to let you know if any k's fit the criteria.

One further point: Algebraic factors are rarely an issue on the Sierp side since 99% of the time they must be for cubes or higher powers. There, it's GFN's to be concerned about but they are much easier to determine. So if algebraic factors concern you, you might stick with the Sierp side.


Gary

[gd_barnes]

Originally Posted by KEP

I've done some testings on Riesel base 633 with a conjectured k=1004, it has by tonight been taken to n=25K, however I need some help on finding out, weather or not the k=64, has to be excluded from further testing (beyond n=25K) or it has to remain in the k's remaining list. Srsieve raised a warning, that k=64 for Riesel base 633, has algebraric factors, however I've never gotten the website used to testing for algebraric factors to work. So can either Gary or Ian tell me weather or not it is composit for all n's for k=64 or it in fact is expected to have a prime eventually?

Btw, consider it reserved, to n=25K, if I want to take it higher I'll let you all know

Regards

KEP

I missed this original post that brought up all of this.

No, you cannot eliminate the k. You can only eliminate even n-values and n-values that are n==(0 mod 3) since the k is both a perfect square and cube.

The same thing applies for k=64 on any Riesel base if the base is not a perfect square or cube.

For 64*633^n-1, as previously shown by David, after eliminating the above for algebraic factors, you're left with n==(1 or 5 mod 6) and after eliminating a factor of 17, you're left with n==(7 or 11 mod 12). There are no other small and consistent factors for those. Therefore, 1 out of every 6 values remains that doesn't have a small factor and hence could contain a prime at higher n-values.

On another note: Have you acquired a lot more machines lately? You have a tremendous amount of work reserved on base 3. Then there's huge bases 63 and 955 and you're still reserving more work. How is base 63 going?


Gary

[KEP]

Base 3, is going to be completed at the end of march for both the reservations, maybe april. Sierp base 63, is still going, but it is currently only running on the dual core. My recent, base 633 reservation, is not goin to be taken further than n=25K, which it completes tonight. Untill my base 3 reservations complete, the quad will only work on the Riesel base 3 tests on all four cores. Currently less than 1500 k's is remaining at a test depth of n=~40-60K. I have not bought new machines, and I may admit, that the k=3677878 to 1G reservation, I would very much like that to be changed back to n=1M. Once I complete S63, R3, aswell as the current R633, then my focus will move to only s955, which I tend to keep till it goes below 1,000 k's remaining. S955 is currently at n=2500, and 8973 k's remaining. I'm not sure if the s63 reservation can be completely finish by the end of march, but in 20-25 days the k=3677878 for R3, will have been taken to n=1M and then complete my original reservation.

But to sum up, all bases is being worked on, even though I still only has 1 DualCore and 1 QuadCore. But on a sidenote, as soon as All R3 and S63 is completed, then all six cores will be hammered on the S955 conjecture.

Hope I got it all, and that it was clear what I stated

Regards

Kenneth

Ps. The reason Riesel base 3 k<700M range is tested to n=40K-60K is because it reached optimal sievedepth for n<=60K, while it was only tested to n=40K.

[MyDogBuster]

Riesel Base 864
Conjectured k = 174
Covering Set = 5, 173
Trivial Factors k == 1 mod 863(863)

Found Primes: 162k's File attached

Remaining k's: Tested to n=25K
4*864^n-1 <------ Proven composite by partial algebraic factors
6*864^n-1
9*864^n-1 <------ Proven composite by partial algebraic factors
49*864^n-1 <------ Proven composite by partial algebraic factors
64*864^n-1 <------ Proven composite by partial algebraic factors
96*864^n-1
114*864^n-1
134*864^n-1
144*864^n-1 <------ Proven composite by partial algebraic factors
169*864^n-1 <------ Proven composite by partial algebraic factors

Base Released