Bases 33-100 reservations/statuses/primes

[mdettweiler]

Ah, I see. I guess the 31 k's just looked like a small amount when they were all listed right in a row on the Riesel base 37 status web page.

Though I definitely don't have any reasonable chance of proving Riesel base 37 in years, I'm still holding out hope that I'll knock out one or two of the k's somewhere in this 10K-20K range.

[KEP]

@Gary:

I think I've finally gotten what you say about proving the conjectures, or at least I've begun understanding. I did some "4 fun" experimentation on Sierp base 7, and it seems to reduce for every bit the amount of k's remaining with ~18.7%, this means that the n-value has to go to between 2^49 and 2^51 before this base is likely to be proven. So I guess for one (me) at least all your explanaition has not been in vain

KEP

[henryzz]

i too have been thinking wrong about this sort of thing
your post has helped massively thanks gary

[mdettweiler]

Originally Posted by mdettweiler

Ah, I see. I guess the 31 k's just looked like a small amount when they were all listed right in a row on the Riesel base 37 status web page.

Though I definitely don't have any reasonable chance of proving Riesel base 37 in years, I'm still holding out hope that I'll knock out one or two of the k's somewhere in this 10K-20K range.

Well, I seem to have struck gold on Riesel base 37 right from the get-go, with two primes in relatively short succession (though, in all fairness, this is not entirely unexpected, based on Gary's calculations above):

672*37^11436-1 is prime!
7466*37^11942-1 is prime!

Max

Edit: Oh, I forgot to mention, these primes were found PRP with Phrot and confirmed prime with a N+1 test via PFGW.

[gd_barnes]

Originally Posted by mdettweiler

Well, I seem to have struck gold on Riesel base 37 right from the get-go, with two primes in relatively short succession (though, in all fairness, this is not entirely unexpected, based on Gary's calculations above):

672*37^11436-1 is prime!
7466*37^11942-1 is prime!

Max

Edit: Oh, I forgot to mention, these primes were found PRP with Phrot and confirmed prime with a N+1 test via PFGW.

Those calculations were for base 3 not base 31 so do not apply in any manner here. They were only to make a point about a very prime base. But the same TYPE of calculation can be used for any base so here we go...

This is better than expected on a non-prime base like 37. For Riesel base 37, there was a 22.5% reduction in k's remaining on a tripling of n-value from n=3333 to 10K. Approximate calculation for this base:

n=3333; 40 k's remaining
n=10K; 31 k's remaining (22.5% reduction on tripling of n-value)
n=30K; 24 k's remaining (22.5% reduction on tripling of n=value)

Breaking it down further:
n=10K; 31 k's remain
n=11.5K; 30.0 remain
n=13.2K; 29.1 remain
n=15.1K; 28.2
17.3K; 27.3
19.9; 26.4
22.8; 25.6
26.2; 24.8
30; 24

Therefore assuming you've testing to around n=12K, I would have expected you to find about one prime by now. Alas, you may find WAY more than expectation or way less and still be within statistical deviations from the norm. Also, it's possible that the primes found from n=3333 to 10K may have been well below or above expectation. It would take a further study over a longer n-range to get a more accurate estimation.

Also, there would be a higher-math method of determing almost exactly how many k's there should be remaining at each n-level based on the density of factors and/or the candidates remaining after sieving to certain depths.


Gary

[mdettweiler]

I get it now. I was under the assumption that your earlier base 3 calculations could be generalized, but now that I re-think that it that doesn't make too much sense.

BTW, found another one last night:

498*37^15332-1 is prime!

[gd_barnes]

Originally Posted by mdettweiler

I get it now. I was under the assumption that your earlier base 3 calculations could be generalized, but now that I re-think that it that doesn't make too much sense.

BTW, found another one last night:

498*37^15332-1 is prime!

Yeah, each base has a different level of 'primeness' so to speak. While base 3 may remove about half it's k's for each ~3-fold increase in the n-value, base 37 clearly removes far less. It has to do with the distribution of small factors.

Is your current search limit at n=~15.3K or so?

If so, your 3 primes are exactly on target with my above estimation, i.e. 28.2 k's remaining (vs. 28 actual) at n=15.1K.

Also, based on the estimate, I would expect a total of 5 primes for n=10K-20K, although it is an extremely rough estimate because only a very small n-range was used in the determination.


Gary

[mdettweiler]

Originally Posted by gd_barnes

Yeah, each base has a different level of 'primeness' so to speak. While base 3 may remove about half it's k's for each ~3-fold increase in the n-value, base 37 clearly removes far less. It has to do with the distribution of small factors.

Is your current search limit at n=~15.3K or so?

If so, your 3 primes are exactly on target with my above estimation, i.e. 28.2 k's remaining (vs. 28 actual) at n=15.1K.

Also, based on the estimate, I would expect a total of 5 primes for n=10K-20K, although it is an extremely rough estimate because only a very small n-range was used in the determination.


Gary

First of all, another prime:

1958*37^16027-1 is prime!

My search limit is now n=16.7K, with no new primes since the n=16027 one above.

[kar_bon]

new PRPs over weekend:

17752 6763
96580 6765
77660 6766
25684 6787
131434 6799
35162 6800
270572 6800
183574 6817
136712 6844
44936 6862
8380 6879
60680 6882
279590 6896
231340 6921
113368 6927
82802 6938
111170 6946
113276 6948
11540 6954
283480 7067
165226 7099
118114 7101
171034 7103

now at n=7118 with 4.99M candidates left, sieved to 5.2G

[kar_bon]

Originally Posted by gd_barnes

Also, it's possible that the primes found from n=3333 to 10K may have been well below or above expectation. It would take a further study over a longer n-range to get a more accurate estimation.

Also, there would be a higher-math method of determing almost exactly how many k's there should be remaining at each n-level based on the density of factors and/or the candidates remaining after sieving to certain depths.

perhaps these info can help, too:
i tested in a few minutes my scripts with Riesel base 37 and got these info:

all 3885 even values for k=2 to 7770 tested
n=1: 688 primes found (3197 remain)
k==1 mod 3: 1295 deleted (1902 remain)
n=2: 414 primes (1488 remain)
3: 277 (1211)
4: 162 (1049)
5: 106 (943)
6: 91 (852)
7: 77 (775)
8: 65 (710)
9: 36 (674)
10: 42 (632)
11: 48 (584)
12: 35 (549)
13: 21 (528)
14: 23 (505)
15: 17 (488)
16: 20 (468)
17: 20 (448)
18: 20 (428)
19: 13 (415)
20: 13 (402)
and further PRPs found:
10,10,14,9,6,2,8,7,7,15,6,10,5,6,5,3,6,6,8,4,4,5,6,2,5,0,2,3,5,3
220 k's remain after n=50 tested
202 after n=60
182 after n=70
169 after n=80
159 after n=90
151 after n=100
116 after n=200
102 after n=300
85 after n=400
79 after n=500
74 after n=600
71 after n=700
64 after n=800
63 after n=900
62 after n=1000

i will try to modify my scripts, because the log file with candidates left / primes/PRP's found isn't looking good!

[mdettweiler]

Riesel base 37 complete to n=20K, four primes in range 10K-20K already reported. Results for 10K-20K have been emailed to Gary.

Edit: Oh, I forgot to mention, I'm releasing this base now.