[QUOTE=gd_barnes;263152]I think you should do base 280...both sides for good measure. :smile:[/QUOTE]

Scripting R63 to n=2000 took a full month on a fast octacore machine. So I'll leave any side of base 280 to somebody considerably more crazy than I am. :smile:

For comparison purposes I tested R3 k<1M n<=10k
R3 has 8 k's remaining at n=10K.
There were 12 primes from n=3K-10K.
So there were 20 k's remaining at n=3K.
Removal rate = 12 / 20 = 60.00% (or 40.00% remaining)

R3 is a lot better than both S63 and R51.
Does this property have any connection with CK? Do bases with a high CK often have high reduction rates?

Edit:
Maybe of interest:
[code]
R3 has 9 k's remaining at n=100.
There were 33 primes from n=30-100.
So there were 42 k's remaining at n=30.
Removal rate = 33 / 42 = 78.57% (or 21.43% remaining)

R51 has 150 k's remaining at n=100
There were 179 primes from n=30-100.
So there were 329 k's remaining at n=30.
Removal rate = 179 / 329 = 54.41% (or 45.59% remaining)

S63 has 170 k's remaining at n=100
There were 249 primes from n=3K-100.
So there were 419 k's remaining at n=30.
Removal rate = 249 / 419 = 59.43% (or 40.57% remaining)[/code]

[QUOTE=henryzz;263207]For comparison purposes I tested R3 k<1M n<=10k
R3 has 8 k's remaining at n=10K.
There were 12 primes from n=3K-10K.
So there were 20 k's remaining at n=3K.
Removal rate = 12 / 20 = 60.00% (or 40.00% remaining)

R3 is a lot better than both S63 and R51.
Does this property have any connection with CK? Do bases with a high CK often have high reduction rates?

[snip][/QUOTE]

There is quite a bit of correlation between "primeness" of the base and CK but there are wide variations. For instance base 280 is not a very prime base yet it has the highest CK. If you think about it logically, the relatively high correlation has to be the case. If there are primes for a high percentage of k's, it means there are few small factors and hence it's difficult for a smaller k to be the conjecture. Since base 3 has so few small factors, by default it must have a high CK.

It has already been stated and known for quite a while that base 3 is by far the most prime base of all of them yet there are several bases with higher conjectures. That and the fact that it is the 2nd lowest base makes it somewhat interesting to me. For R3 with a CK=~50G, I would expect ~100,000 k's remaining at n=100K and ~15,000 k's remaining at n=1M. That's still a whole lot of k's but the fact that < 1 in 1 million k's remain at n=1M (a total of ~25G k's to search) is likely unique to only base 3. It would be the equivalent of proving a base with a CK of 1,000,000!

R39 is now complete to n=25000. 251 primes found between n=20000 and n=25000.

I am releasing this conjecture.

Gary, I will e-mail you the updated sieve files.

[QUOTE=rogue;263675]R39 is now complete to n=25000. 251 primes found between n=20000 and n=25000.

I am releasing this conjecture.

Gary, I will e-mail you the updated sieve files.[/QUOTE]

Nice work Mark. There are officially 3675 k's remaining at n=25K for R39.

I would like to reserve S61 to n=100K

S46 is complete to n=100K; 6 primes found for n=50K-100K shown below; 21 k's remain; base released.

Primes:
[code]
13488*46^60373+1
10776*46^69160+1
7333*46^69526+1
12643*46^83494+1
6391*46^98093+1
13443*46^99244+1
[/code]

R36 is complete to n=100K; an [I]amazing[/I] 29 primes were found for n=50K-100K shown below; base released. Officially there are 49 k's remaining at n=100K but subtracting the 2 k's previously found for n>100K leaves 47 k's remaining for the base.

Primes:
[code]
57785*36^50056-1
62087*36^50861-1
92908*36^51559-1
47927*36^51560-1
37480*36^53038-1
60029*36^53684-1
54759*36^57184-1
46102*36^59802-1
57462*36^60549-1
50183*36^62453-1
41727*36^62674-1
75703*36^64193-1
51172*36^65003-1
16168*36^66422-1
11014*36^66423-1
41669*36^68136-1
29847*36^70763-1
48950*36^71618-1
15909*36^72122-1
95350*36^73163-1
86618*36^76585-1
15687*36^78908-1
39898*36^81121-1
49663*36^83542-1
98530*36^90816-1
110423*36^92240-1
32302*36^93267-1
25679*36^98885-1
40327*36^99249-1
[/code]

Quite an impressive amount of primes. I would like to do some testing myself but it is too much work for me to make an efficient impact. Anyone fancy a team effort? There should be lots of primes to find.

Some stats:
Sieving upto 200k with the 46ks not that far would leave ~120k candidates. The average testing time(based on 160k) would be about 1000 seconds(16 minutes) on a Athlon II @2.5 GHz. That would mean about 33000 cpu hours for testing. Hopefully the 2/3 remaining after doubling n would continue and we would find ~16 primes ~half of which would be top 5000. A quick spreadsheet gives a probably inaccurate regression line prediction of ~800k for proof. Anyone interested?

edit: forget the testing times. I forgot to use a k value(used k=1). The testing time will be greatly increased due to higher fft length.

[QUOTE=henryzz;265398]Quite an impressive amount of primes. I would like to do some testing myself but it is too much work for me to make an efficient impact. Anyone fancy a team effort? There should be lots of primes to find.

Some stats:
Sieving upto 200k with the 46ks not that far would leave ~120k candidates. The average testing time(based on 160k) would be about 1000 seconds(16 minutes) on a Athlon II @2.5 GHz. That would mean about 33000 cpu hours for testing. Hopefully the 2/3 remaining after doubling n would continue and we would find ~16 primes ~half of which would be top 5000. A quick spreadsheet gives a probably inaccurate regression line prediction of ~800k for proof. Anyone interested?

edit: forget the testing times. I forgot to use a k value(used k=1). The testing time will be greatly increased due to higher fft length.[/QUOTE]

?? You should take a little more time before responding.

First off, we have enough team efforts. Secondly, where are you getting n=800K for a possible proof? That's not possible for 3 reasons: (1) The k's remaining become much lower average weight as we go. (2) n=50K-100K was an anomoly. For n=25K-50K, the reduction was only from 98 k's to 78 k's. That was low but the percentage reduction for n=50K-100K was high. It would be better to use an avg. of the two, i.e. from 98 to 49 k's for a quadrupling of the n-range for n=25K-100K. (3) Your regression has been calculated incorrectly. Assuming a constant reduction to 2/3rds for every doubling of the n-range, it would be: 47*2/3=31 k's remaining at n=200K, 31*2/3=21 k's at n=400K, 21*2/3=14 k's at n=800K, 14*2/3=9 k's at n=1.6M, 9*2/3=6 k's remaining at n=3.2M, etc.. Even if the reduction was constant and was that high to begin with (which it isn't due to the anomoly), the proof clearly would be at n>10M.

This will not be proven in most of our lifetimes even assuming increases in computer speed without a new way of calculating prime numbers. I expect its actual proof to be at n>50M.

All of this said, it is a nice base and believe-it-or-not should eventually be proven much more quickly than most bases with a conjecture of k>100K.

BTW there are 47 k's remaining vs. 46.

S42 is complete to n=100K; 5 primes found for n=50K-100K shown below; 27 k's remain; base released.

Primes:
[code]
3046*42^50828+1
2903*42^53046+1
12041*42^55780+1
8382*42^59634+1
6236*42^81124+1
[/code]