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Bases 33-100 reservations/statuses/primes

1 Attachment(s)

[quote=gd_barnes;137054]
I hope that no new bases are started for the next several months now! :smile:

Gary[/quote]

I have already finished this one.

Willem.

 gd_barnes 2008-07-12 08:45

bases 33-100

Willem.[/quote]

Willem,

I forgot to mention here when I checked this a week ago: Nice work on proving the Riesel base 33 conjecture at k=764. All k-values were accounted for with 2 of them having algebraic factors. The proof was added to the web pages after I checked it.

To all: If you want to tackle a base or two higher than 32, that's fine as long as the conjecture is low; preferrably < ~2500. Most low conjectures are easy to administer and check.

You'll have to determine what the lowest Riesel or Sierp value is with a covering set. There's a thread that talks about software that can do this. In the past, I've used a crude method that uses srsieve sieving software that is quite accurate for determining low conjectures but it's a little combersome to set up and use.

Before starting on a base, please let us know that you are reserving it, what the conjectured value is, and what the covering set is.

Gary

[QUOTE=gd_barnes;137693]Before starting on a base, please let us know that you are reserving it, what the conjectured value is, and what the covering set is.

Gary[/QUOTE]

Ah yes, other bases. I wrote a program to calculate conjectures myself. I've also done some more work on the Riesel bases until 50.
Bases 34, 38, 41, 43, 44, 47 and 50 are trivial to prove.
Bases 35, 39, and 40 have conjectures higher than a million.
Bases 36, 37, 42, 45, 46, 48 have between 20 and 100 remaining k at n = 10,000
Base 49 has 1 remaining k. That one I'd like reserved for myself.

Gary, I hadn't mentioned this because I did't want to hand over the data to you and dropping you in a hole like I did with the base 25. I'll post the trivial conjectures when I have them in the same format as the one that I gave last week.
As for the others, tell me how you like the data and I'll format it that way.

Willem.

base 34

Base 34 has riesel = 6.
If (k mod 3) == 1 then for any n there is a factor 3. This eliminates k = 1 and k = 4.

k n
2 1
3 1
5 2
6 Riesel

covering set {5, 7}
odd n: 7
even n: 5

base 38

Base 38 has Riesel = 13
Covering set {3. 5. 17}

k n
1 1
2 2
3 1
4 1
5 2
6 1
7 7
8 2
9 43
10 1
11 766
12 2
13 Riesel

Listed by decreasing n:
k n
11 766
9 43
7 7
5 2
2 2
12 2
8 2
3 1
1 1
6 1
10 1
4 1

Base 41

Base 41 has Riesel = 8
If k is odd then any n will have a factor 2. This eliminates k = 1, 3, ,5 , 7.
If (k mod 5) == 1 then any n will have a factor 5. This eliminates k = 6.

k n
2 2
4 1
8 Riesel
Covering set {3, 7}

base 43

1 Attachment(s)
Base 43 has Riesel = 672
covering set = {5, 11, 37}

Excluded:
k mod 2 = 1
k mod 3 = 1
k mod 7 = 1

highest primes
308 624
12 203
516 202
450 162
494 148
476 101
104 77
560 70
384 48
188 37

All primes in attachment.

base 44

Base 44 hase Riesel = 4
Covering set = {3, 5}

Primes
k n
1 1
2 4
3 1

base 47

Base 47 has Riesel = 14
Covering set = {3, 5 ,13}

excluded: odd k

k n
2 4
4 1555
6 1
8 32
10 51
12 1

base 50

Base 50 has riesel = 16
covering set = {3, 17}

(k mod 7) == 1 has factor 7. This eliminates 1, 8 and 15

k n
2 2
3 1
4 1
5 12
6 6
7 1
9 1
10 1
11 6
12 1
13 19
14 66

Listed by decreasing n:
14 66
13 19
5 12
11 6
6 6
2 2
12 1
4 1
3 1
9 1
10 1
7 1

 gd_barnes 2008-07-12 19:42

Bases 33-100 reservations/statuses

Report all reservations/status for bases 33-100 in this thread.

These would be low priority and something to be worked on 'just for fun' that are not too CPU-intensive. It is preferred that you only stick with low-conjectured bases; preferrably with a conjecture of k<2500.

Before starting on a base, please state that you are reserving it, its conjectured value, and the covering set.

When reporting things, please account for all k-values in some manner. This will help keep the admin. effort to a minimum. :smile:

Gary

 gd_barnes 2008-07-12 19:59

[quote=Siemelink;137696]Ah yes, other bases. I wrote a program to calculate conjectures myself. I've also done some more work on the Riesel bases until 50.
Bases 34, 38, 41, 43, 44, 47 and 50 are trivial to prove.
Bases 35, 39, and 40 have conjectures higher than a million.
Bases 36, 37, 42, 45, 46, 48 have between 20 and 100 remaining k at n = 10,000
Base 49 has 1 remaining k. That one I'd like reserved for myself.

Gary, I hadn't mentioned this because I did't want to hand over the data to you and dropping you in a hole like I did with the base 25. I'll post the trivial conjectures when I have them in the same format as the one that I gave last week.
As for the others, tell me how you like the data and I'll format it that way.

Willem.[/quote]

The format that you did base 33 in was a good one. All k's were accounted for. So if the conjectured-k is not too big, send me a file or spreadsheet of an accounting of all k-values. You don't have to list all that have trivial factors and algebraic factors but a statement as to what those are helps.

In other words, it's best if everything is sorted by k except for the trivial k's. Example on some fictional base:

[code]
k==(1 mod 3) is trivial

k prime/status
2 5
3 remaining
5 1
6 3
8 2
9 algebraic factors
11 remaining
etc. up to the conjectured k-value
[/code]

This accounts for everything in a nutshell. Alternatively, if the algebraic factors are very consistent (which they frequently are not like on base 33), you can just state something like "k's that are a perfect square have algebraic factors" and not show those in the list of k's.

Gary

base 49 PRPs

1394*49^52698-1
1266*49^36191-1
230*49^24824-1
1706*49^16337-1
1784*49^13480-1
786*49^6393-1

I am running PFGW on these at the moment, the confirmation will follow later.

Willem.

 gd_barnes 2008-07-13 13:24

1266*49^36191-1
230*49^24824-1
1706*49^16337-1
1784*49^13480-1
786*49^6393-1

I am running PFGW on these at the moment, the confirmation will follow later.

Willem.[/quote]

Wow; you've put in some serious work on this one! It looks like you noticed that I did a preliminary search on the base as a check like I do on all of them. Usually I only go to n=2K but I couldn't believe that there were still so many k's remaining so I went to n=5K thinking I might have the wrong conjectured value. I was going to post a note questioning you only having one k-value remaining when I still had 7 remaining at n=5K but I see that indeed you've knocked them all out except one. That's some serious CPU crunching there to get such a high base past n=50K!

It looks like you may be in top-5000 territory (i.e. n>60K) on the last k. Good luck with it! :smile:

Gary

[QUOTE=gd_barnes;137740]Wow; you've put in some serious work on this one! It looks like you noticed that I did a preliminary search on the base as a check like I do on all of them.
It looks like you may be in top-5000 territory (i.e. n>60K) on the last k. Good luck with it! :smile:

Gary[/QUOTE]

I could have known that a casually mentioned figure would be picked up by you. No half baked entries on your pages!
By now the six primes were confirmed by PFGW.

Willem.

Riesel base 48

1 Attachment(s)
The riesel conjecture for base 48 = 4208, with cover set {7, 13, 37, 61}
Even => 7
6m+1 => 13
6m+3 => 37
6m+5 => 61

checked n upto 10000
total k 4117
total p 4043
Remaining k 74

I've checked the 4043 primes with pfgw, they all hold up. Of the remaining k, two are squares but I couldn't eliminate them. There is one k that can be divided by 48. but I coudn't eliminate that one either.

Top ten primes
1422 9235
3179 9107
1021 8570
4108 8296
3382 7927
1103 7918
475 7424
2449 7244
3907 7083
3541 7078

All the k's and primes are in the attachment. Feel free to find more primes.

Enjoy, Willem.

Riesel base 39 & 40

The lowest Riesel for base 39 = 1,352,534, with covering set {5, 7, 223, 1483}.
The lowest Riesel for base 40 = 3,386,517, with covering set {7, 41, 223, 547}.

I've calculated these with my riesel generator, but I didn't generate any k. If you feel like generating a lot of primes, here is your chance.

Willem.

 gd_barnes 2008-07-20 19:55

Thanks Willem. Your base 48 info. is exactly what we need on a new base...covering set and all. :smile:

One thing I'll add for everyone's reference: Willem has correctly removed all k==(1 mod 47) remaining, which have a trivial factor of...you guessed it...47.

Gary

 gd_barnes 2008-07-21 11:19

Willem,

What is your search limit on k=2186 for Sierp base 49 and how high were you going to take it? I have used n=5K because that was how high I searched it to get all of the small primes for the base. I assume you've searched it somewhere above n=50K since you have a prime for k=1394 at n=52698.

Thanks,
Gary

[QUOTE=gd_barnes;138102]Willem,

What is your search limit on k=2186 for Sierp base 49 and how high were you going to take it? I have used n=5K because that was how high I searched it to get all of the small primes for the base. I assume you've searched it somewhere above n=50K since you have a prime for k=1394 at n=52698.

Thanks,
Gary[/QUOTE]

My Riesel 49 effort is at 88,000 and continuing until 100,000. After that I'll see.

Willem.

Riesel base 46

1 Attachment(s)
Hi everyone,

here is my effort on Riesel base 46. The conjectured lowest riesel is 8177. The cover set is {29, 47, 73}. While generating the k's I've ignored even k's and k mod 23 = 1. At n = 10,000 there are 22 k's left:
93
800
870
1317
1362
2819
3147
3194
3383
3812
4419
4580
5940
6060
6062
6297
7157
7284
7424
7472
7520
7848

I've checked against squares, there are none left. k = 4278 = 93 *48 and 93 is still in the list. That allows me te remove k = 4278.

The top ten of primes is:
6224 8837
4464 7100
3504 4377
6524 3504
7715 3482
1940 3473
5979 3275
2042 3010
4610 2724
6263 2372

All the primes have been tested with pfgw and are attached. Find some more!

Willem.

 gd_barnes 2008-07-25 07:02

here is my effort on Riesel base 46. The conjectured lowest riesel is 8177. The cover set is {29, 47, 73}. While generating the k's I've ignored even k's and k mod 23 = 1.[/quote]

I assume that you meant that you ignored k==(1 mod 3) and (1 mod 5).

Ignoring k==(1 mod 23) would be for Riesel base 47. We would never ignore even k's.

Gary

Riesel base 45

1 Attachment(s)
Hi everyone, here is my base 45 effort. The Riesel conjecture is 22564.
I've taken this to n = 10,000. while ignoring odd k and (k mod 11) = 1.
As 1080 = 24*35 and 16740 = 372*45 I've left hem out.
This leaves the 22 following k:
24
372
1264
1312
2500
2804
4210
4484
5128
6094
6372
7246
10096
10518
12950
13456
13548
15432
17918
19252
20654
21274

There are 2 squares remaining, 2500 and 13456, but that is just coincidence I think. There are many square k's that do have a prime.
Also, 24 = 6*2*2. Several of the conjectures remove k = 6*square, but I don't understand why. How can I check if it can be removed here also?

The top ten of primes:
13546 9069
17734 8019
19102 7368
14324 7281
9938 7240
4628 7209
4622 7116
6554 6462
2230 5892
7750 4586

I've attached and doublechecked all the primes that I found.

Willem.

 gd_barnes 2008-08-03 12:05

There are 2 squares remaining, 2500 and 13456, but that is just coincidence I think. There are many square k's that do have a prime.
Also, 24 = 6*2*2. Several of the conjectures remove k = 6*square, but I don't understand why. How can I check if it can be removed here also?
Willem.[/quote]

It's not always very easy. The way I do it is to look for patterns in the factors of the various n-values for specific k-values. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.:
11*13
179*181
etc.

In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors. That's what I ran into on base 24.

For your 3 cases here, you have:
k=24:
n-value : factors
1 : 13*83
2 : 23*2113
3 : 17*103*1249
4 : 23*163*26251
9 : 2843*6387736694293
Analysis:
For n=3 & 4, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors.
For n=9, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point.

k=2500:
n-value : factors
1 : 19*31*191
2 : 13*173*2251
3 : 89*2559691
4 : 19*73^2*103*983
9 : 9439*4280051*46824991
For n=9 same explanation as k=24.

k=13456:
n-value : factors
1 : 269*2251
2 : 17*23*227*307
3 : 31*39554129
4 : 7*23*467*503*1459
7 : 3319*1514943103721
For n=7 same explanation as k=24.

The prime factors for n=9, n=9, and n=7 respectively make it clear to me that these k-values should all yield primes at some point so you are correct to include them as remaining.

The higher-math folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of n-values on them but not for all of the n-values. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime.

Gary

base 42 Riesel

1 Attachment(s)
Hi everyone,

here is my work on the base 42 Riesel conjecture 15137. The covering set is {5, 42, 353}.
After taking n to 10,000 there were 72 k left. I removed 2058 = 49 * 42 as 49 is still in the list of k.
There are also 5 squares in the list (49, 1369, 2304, 3721 and 10201), but they give no obvious deductions.

This leaves:
[code]
49
386
603
1049
1160
1426
1633
1678
2304
2464
2538
2753
3428
3734
4299
4903
5118
5417
5677
5820
5899
5978
6333
6623
6664
6836
6838
6964
7016
7051
7309
7489
7614
7658
7698
7913
8297
8341
8384
8453
8524
9029
9201
9418
9633
9848
10026
10114
10276
10663
10923
11052
11267
11781
11911
11996
12039
12125
12127
12151
12213
12598
13288
13329
13347
13425
13632
13757
13898
14576
15024
[/code]

Enjoy, Willem.

 gd_barnes 2008-09-06 07:56

here is my work on the base 42 Riesel conjecture 15137. The covering set is {5, 42, 353}.
After taking n to 10,000 there were 72 k left. I removed 2058 = 49 * 42 as 49 is still in the list of k.
There are also 5 squares in the list (49, 1369, 2304, 3721 and 10201), but they give no obvious deductions.

Enjoy, Willem.[/quote]

Looks good. Nice work. Actually only 2 of your squares are remaining: k=49 and k=2304. As you showed in your list, k=1369 has a prime at n=7577, k=3721 has a prime at n=4611, and k=10201 has a prime at n=2129.

Gary

Riesel base 37

1 Attachment(s)
Hi there Gary, thanks for clarifying my muddled statement.

here is the next one, base 37. The conjecture is 7772, with set = {5, 19, 137}. At n = 10,000 there are 30 k remaining:
[code]
284
498
522
590
672
816
1008
1578
1614
1842
1958
2148
2606
2640
3336
3480
3972
4356
4428
4542
4806
5262
5376
5910
5946
6288
6752
6792
7088
7352
7466
[/code]

I've attached the primes found. Here is the list of the highest primes:
7058 8314
1334 7883
5156 7797
6480 7763
554 7472
7124 6396
474 3952
998 3572
912 3394
1956 3250

Willem.

Riesel base 35

1 Attachment(s)
Hi everyone,

I've double checked my work on the Riesel 35 Conjecture. The value is 287860. I've taken the k's to n=200 with PFGw. Then I sorted out the multiples of 35 and the squares. ThenI took the remainder to n=5000
I've attached the 1559 k's that I had left.
The top ten primes list is this:
65216 4986
248264 4980
104690 4978
126050 4978
286652 4976
129052 4975
229454 4974
48772 4965
169448 4964
7874 4962

All the primes are in a file that zips to 350k, that doesn't fit on the forum. The can be sent if so desired.

Willem.

 gd_barnes 2008-10-08 06:32

I've double checked my work on the Riesel 35 Conjecture. The value is 287860. I've taken the k's to n=200 with PFGw. Then I sorted out the multiples of 35 and the squares. ThenI took the remainder to n=5000
I've attached the 1559 k's that I had left.
The top ten primes list is this:
65216 4986
248264 4980
104690 4978
126050 4978
286652 4976
129052 4975
229454 4974
48772 4965
169448 4964
7874 4962

All the primes are in a file that zips to 350k, that doesn't fit on the forum. The can be sent if so desired.

Willem.[/quote]

Can you please send the primes to me at: gbarnes017 at gmail dot com

Thanks,
Gary

 kar_bon 2008-10-08 08:05

i give base 35 a few days shot!

so far:
sieved all 1559 (minus k with primes) upto p=402M for n=5k-100k
checked upto n=5249
38 more primes found
6.5M candidates left

sieving further!

will mail primes when more available.

 kar_bon 2008-10-08 08:06

reserved base 35 from n=5k-100k

so Siemelink can check another base :-)

 michaf 2008-10-08 16:21

Rest assured that it'll take you quite some time to take it to 100k...

(Base 31, also not very prime (compared to base 3), now runs erm... 4 or 5 months or so :) )

 gd_barnes 2008-10-08 20:07

[quote=michaf;144873]
Rest assured that it'll take you quite some time to take it to 100k...

(Base 31, also not very prime (compared to base 3), now runs erm... 4 or 5 months or so :) )
[/quote]

To clarify: Actually base 31 is very prime compared to most bases. It is much more prime than base 35 is. But of course nothing compares to base 3. So far, only base 7 comes close. I also suspect base 15 will be quite prime.

It seems that all bases where b=2^q-1 are very prime as compared to their neighbors.

I would expect many CPU years to get base 35 up to n=100K.

Gary

 kar_bon 2008-10-09 08:25

base 35

status for a run over night (one core of Quad):

sieved to p=535M (from 402M) about 90000 candidates less
llr tested upto n=5538 (from 5249) = 20000 pairs tested

31 more primes found = now 1490 k's left (from 1559)

after deletion of these sequences: 6,270,000 candidates all over left (from 6.5M)

to llr-test a candidate it takes about 3.7 s but it's more efficient to find a prime and delete a sequence from the sieve-file (about 5000 candidates for a sequence) as only sieving for now!

Good luck Karsten!

I took base 19 to 30,000. That has a similar amount of remaining k's. It took a long long time. But my PCs are quite old.
Anyway, when I find the time I'll publish the numbers on base 36 as well. I've assigned a core to that, it is up to 22,000 by now, 100 k's left or so.

Willem.

 robert44444uk 2008-10-15 13:33

Just catching up with this thread. FYI, "covering.exe" had an error in it, which invalidated the conjectured lowest Riesel b=43. The error has been corrected but you should get the latest version of covering.exe if you intend further work on higher bases.

I intend to at least list the lowest conjectured covering sets for Sierpinski and Riesel up to b=1000, but the rate these numbers increase by in the power series suggests that very few will be proven, and limitations on bases for prime proving algorithms accentuates. I agree that concentrating on b=3 to 100 is worthwhile.

But the lists may show that the average value of lowest covers decreases, and that is worth looking at.

 KEP 2008-10-15 15:39

[QUOTE=robert44444uk;145447]Just catching up with this thread. FYI, "covering.exe" had an error in it, which invalidated the conjectured lowest Riesel b=43. The error has been corrected but you should get the latest version of covering.exe if you intend further work on higher bases.

I intend to at least list the lowest conjectured covering sets for Sierpinski and Riesel up to b=1000, but the rate these numbers increase by in the power series suggests that very few will be proven, and limitations on bases for prime proving algorithms accentuates. I agree that concentrating on b=3 to 100 is worthwhile.

But the lists may show that the average value of lowest covers decreases, and that is worth looking at.[/QUOTE]

Hi Robert

Could you do me a favor and come up with the lowest conjectured value for all base <=2^10 or <=1024? I've a dream and hope to see all the conjectures for bases<=1024 to be taken to n<=50M or proven before I turn hundred... it gives us almost 74 years to reach that goal and with the help of primegrid I actually think it will be possible to take all those conjectures up to n<=50M in maybe less than 74 years. However it requires new scripts to do the initial testings, and a lot more people to desire to work on producing and storing the primes for later construction of proofs. But you think that you can produce the lowest conjecture value for all bases <=1024 and their covering sets?

Just hope that Gary is up for the extra load of work... anyway I'm most likely going to do something else besides the conjectures for a while, and when (if) I return, I'm gonna see how goes with the base 3 conjectures :smile:

KEP!

1 Attachment(s)
Ah, like this you mean? I conjured this list with my own program. Any checks/errors would be welcome. I used the primes smaller than 10,000 for the covering sets.

Willem.
--
the only one not in the list is the riesel for base 921. It took too long so I pressed ^C

 KEP 2008-10-16 09:24

[QUOTE=Siemelink;145481]Ah, like this you mean? I conjured this list with my own program. Any checks/errors would be welcome. I used the primes smaller than 10,000 for the covering sets.

Willem.
--
the only one not in the list is the riesel for base 921. It took too long so I pressed ^C[/QUOTE]

Any chance the 3 conjectures that states x.xxexx, can be conjectured more exact? Also any chance you can make a list for Sierpinski side too? Also I may add, that even in 74 years it is going to be a tough task to reach a milestone of n<=50M for all the billions of k's that needs to be tested at n>25000, but if popularity is going to grow with this project and computer speeds is going to double every second year as it has up to date, we will be able to get at least a great deal towards that goal :smile:

But let me hear the answers for my questions, then we can always discuss future milestones... maybe we also should discuss milestones with a shorter lifeexpectancy :smile:

Regards

KEP

 MrOzzy 2008-10-16 09:51

[quote=Siemelink;145481]Ah, like this you mean? I conjured this list with my own program. Any checks/errors would be welcome. I used the primes smaller than 10,000 for the covering sets.

Willem.
--
the only one not in the list is the riesel for base 921. It took too long so I pressed ^C[/quote]

I've found a better conjecture for Riesel base 71: 1132052528 in stead of 80375729488 like it is listed in the file you supplied. I don't know how to prove the conjecture, I just use the tools I know of. I've checked the factors for all n of base 71 upto n=100. The factors making all results composite are:

3 for all odd n.
13,37,73,109,1657 and 2521 for all even n.

In case I missed something these are the primes considered by covering.exe: 3,5113,2521,1657,37,13,1954357,17,113,577,19,73,109,282439,87553,3889,501841,937,1297,180001

Edit: 75070204388 is also smaller as 80375729488 and has a full covering set, still 1132052528 is better :-)

 kar_bon 2008-10-16 09:57

Y.Gallot stated in the paper (don't know were to find) "ON THE NUMBER OF PRIMES IN A SEQUENCE" in 2001 for the Sierpinski conjecture:
[quote]
We have a 50% chance of solving Sierpinski problem at N = 2^43 about 10^13.
We have a 5% chance of solving it at N = 2^30 about 10^9.
We have a 95% chance of solving it at N = 2^81 about 10^24.
Note also that the chances at 2^20, 2^21 and 2^22 are respectively about 10^-6, 10^-5 and 10^-4.
[/quote]

and for the Riesel problem he stated:
[quote]
We have a 50% chance of solving Riesel problem at N = 2^70 about 10^21.
We have a 5% chance of solving it at N = 2^47 about 10^14.
We have a 95% chance of solving it at N = 2^134 about 10^40.
Note also that the chances at 2^20, 2^25 and 2^30 are respectively about 10^-40, 10^-16 and 10^-8.
[/quote]

the smallest not tested exponents for Riesel are at about n=2.3M (so about 2^21)!

just an idea of time for only [b]one[/b] conjecture!

 MrOzzy 2008-10-16 10:06

I also found a better solution for Riesel base 66:

101954772 (was 144915105) with covering set:
67 for even n
7,17,37,73,613 for odd n.

In case I missed something, here are the primes considered: 67,4423,4357,7,613,17,409,2729,19,109,37,512713,37057,73,15217,14653,97,60289,937,1153

 robert44444uk 2008-10-16 10:28

I am happy to post the Sierpinski side but I need to organise my results, which are considering primes up to 100000 and up to 144-cover.

I can also check the work done on the Riesel side up to similar limits. i am using the revised covering.exe program so it will be interesting to see if I come up with alternative values! I will do this before doing the Sierpinski.

[QUOTE=MrOzzy;145536]I've found a better conjecture for Riesel base 71: 1132052528 in stead of 80375729488 like it is listed in the file you supplied.[/QUOTE]

That's quite possible, I didn't try to find the lowest riesel, only the riesel with the lowest amount of primes.

Willem.

 gd_barnes 2008-10-17 01:24

[quote=KEP;145454]Hi Robert

Could you do me a favor and come up with the lowest conjectured value for all base <=2^10 or <=1024? I've a dream and hope to see all the conjectures for bases<=1024 to be taken to n<=50M or proven before I turn hundred... it gives us almost 74 years to reach that goal and with the help of primegrid I actually think it will be possible to take all those conjectures up to n<=50M in maybe less than 74 years. However it requires new scripts to do the initial testings, and a lot more people to desire to work on producing and storing the primes for later construction of proofs. But you think that you can produce the lowest conjecture value for all bases <=1024 and their covering sets?

Just hope that Gary is up for the extra load of work... anyway I'm most likely going to do something else besides the conjectures for a while, and when (if) I return, I'm gonna see how goes with the base 3 conjectures :smile:

KEP![/quote]

Not possible, even if you assume a doubling of computer speeds every 2 years, which is highly unlikely! You are forgetting that base 1000 for 1000^50000000 is a much larger test than 2^50000000. Even so, taking all k's on Riesel base 2 up to n=50M will be a serious challenge in most of our lifetimes.

Yep, I'll find time to update the web pages. It might be a couple of months before I get all this info. in there but it will get there.

There's one thing that I want to bring up here: Neither I nor anyone else here can claim ownership of these conjectures. This project has only been intended to organize such efforts towards the conjectures not being worked on by others, not own them. We will not be offended if anyone wants to break off and create a separate project for a specific base or two. The base 5 project has made HUGE progress on its own in a few short years on an extremely tough base and personally, I'm glad that they have done it so that we don't need to. The same applies to Sierp base 4.

KEP, you really enjoy base 3 so if you want to create a separate project for it, go right ahead. I'll be glad to assist with that. One guarantee: It's a lot more work to administer these things than you'll ever imagine in the beginning. :smile:

All of that said, if another effort is 'dropping the ball' such as has happened with RieselSieve, you can bet we will step in and pick up the slack if the effort goes dormant for too long and there has been little communication about when it will start up again. But if the effort subsequently 'comes back', we'll gladly let them pick it back up again and communicate any progress made.

Gary

 gd_barnes 2008-10-17 01:34

[quote=kar_bon;145538]Y.Gallot stated in the paper (don't know were to find) "ON THE NUMBER OF PRIMES IN A SEQUENCE" in 2001 for the Sierpinski conjecture:

and for the Riesel problem he stated:

the smallest not tested exponents for Riesel are at about n=2.3M (so about 2^21)!

just an idea of time for only [B]one[/B] conjecture![/quote]

Very interesting odds Karsten! It's even more difficult than what I had attempted to compute in another thread, which had the Riesel base 2 conjecture with a better than 50-50 chance of being solved by n=16T (n=16*10^12). That is, I had computed that there should be < 0.5 of a k remaining at that point.

I've now realized that there's one large error that I made in my computations. I assumed the primes would continue to come at the same exponential reducing rate. That is an incorrect assumption because the k's remaining will have respectively less average weight than the k's where primes have been already found. Therefore I'm sure that Gallot's estimate is far more accurate.

Edit: In KEP's defense here, he did not state that the conjectures needed to be proven to realize his dream; only that they need be tested to n=50M. That seems to be a reasonable goal for Riesel base 2 in most of our lifetimes but not for 2046 bases, i.e. 2 bases each for 2 thru 1024! :-)

Gary

 robert44444uk 2008-10-17 04:49

[QUOTE=R. Gerbicz;145585]Better Riesel values, also up to base=1024:

Saved me the work!

Robert - could you disclose to what cover these are tested. Your program requires every cover to be tested. For example it is possible (but unlikely) for a 5-cover to be smallest.

 robert44444uk 2008-10-17 07:16

Robert G's list produces lower values than Siemelink's as follows:

[code]
66 101954772
71 1132052528
120 166616308
127 93902377422
156 2113322677
175 278467080
195 582483712
238 5415261
240 2952972
280 513613045571841
303 85368
323 93896
325 112882226
345 1295243216
358 27606383
435 31732727570
453 4658266
511 40789000085994
525 8364188
541 15546458
570 12511182
591 30820
661 2518794379382
685 518792
728 212722
777 23485096
796 27199220
799 1885767686976
801 40381102
826 131420459393
855 7419914968008
876 51768432
906 171998037
910 5005381602981
946 2156122023
960 61681833328
963 22349616
966 699327630
981 112303013130
1020 94655888
[/code]

 kar_bon 2008-10-17 09:57

Riesel base 35 status

LLR-tested for all k's upto n=6050
sieved to 3.9B=3.9*10^9
124 primes found so far (from 1559)
5.4M candidates left

 gd_barnes 2008-10-17 10:52

[quote=kar_bon;145653]LLR-tested for all k's upto n=6050
sieved to 3.9B=3.9*10^9
124 primes found so far (from 1559)
5.4M candidates left[/quote]

In order to update the web page, I'll need to get a list of the primes. Otherwise, the highest primes list will be out of sync with the n-range tested.

I'll create a page of the k's remaining at n=5K from Willem's earlier posted list a little later today.

 R. Gerbicz 2008-10-17 15:04

[QUOTE=robert44444uk;145642]Saved me the work!

Robert - could you disclose to what cover these are tested. Your program requires every cover to be tested. For example it is possible (but unlikely) for a 5-cover to be smallest.[/QUOTE]

In fact if you test for example exponent=36, then the program will find all coversets (for the listed primes) not only for period=36 but also for the divisors of 36, so for period=2,3,4,6,9,12,18,36.

As I remember I tested for exponent=144 but for low limit for primes (limit=10000), after it was switched to test exponent=8,24,36,48 for large limit. But today I think it was really unnecessary, here it is a quick stat for the Sierpinski side b=2-1024:
[code]
number of period=2 is 525
number of period=3 is 53
number of period=4 is 224
number of period=6 is 107
number of period=8 is 18
number of period=9 is 2
number of period=12 is 75
number of period=16 is 1
number of period=18 is 2
number of period=24 is 11
number of period=36 is 2
number of period=48 is 1
number of period=72 is 1
number of period=144 is 1
max prime in coverset=731881 at b=855
[/code]
And for Riesel side b=2-1024:
[code]
number of period=2 is 528
number of period=3 is 53
number of period=4 is 230
number of period=6 is 105
number of period=8 is 23
number of period=10 is 2
number of period=12 is 63
number of period=16 is 1
number of period=18 is 1
number of period=24 is 9
number of period=36 is 5
number of period=48 is 1
number of period=144 is 1
max prime in coverset=921601 at b=959
[/code]
Yesterday there was a very large prime in one of the coverset, about 300million, but I found a lower k value for that. Seeing this table it is more than enough to search only for primelimit=1M or 2M.

I would be glad if some of you could find a better k value.

 gd_barnes 2008-10-17 23:01

I've double checked my work on the Riesel 35 Conjecture. The value is 287860. I've taken the k's to n=200 with PFGw. Then I sorted out the multiples of 35 and the squares. ThenI took the remainder to n=5000
I've attached the 1559 k's that I had left.
The top ten primes list is this:
65216 4986
248264 4980
104690 4978
126050 4978
286652 4976
129052 4975
229454 4974
48772 4965
169448 4964
7874 4962

All the primes are in a file that zips to 350k, that doesn't fit on the forum. The can be sent if so desired.

Willem.[/quote]

Willem,

Can you send me all your primes on Riesel base 35?

Thanks,
Gary

 gd_barnes 2008-10-18 02:03

[quote=kar_bon;145653]LLR-tested for all k's upto n=6050
sieved to 3.9B=3.9*10^9
124 primes found so far (from 1559)
5.4M candidates left[/quote]

Karsten,

Per your Email showing primes found up to n=6060, I have now listed all k's remaining and highest primes found for Riesel base 35.

Willem,

Your list of k's remaining for this base had a slight error in it. You had both k=94 and 115150 remaining. Since 115150=94*35^2, it can be eliminated. The list of primes that you found will help me do a little more verification.

Karsten,

You can eliminate k=115150 from your testing. The web pages reflect the removal. Once you do that, you might check your file to verify that there are 1434 k's remaining at n=6060.

Gary

 robert44444uk 2008-10-18 12:35

[QUOTE=R. Gerbicz;145674]
Yesterday there was a very large prime in one of the coverset, about 300million, but I found a lower k value for that. Seeing this table it is more than enough to search only for primelimit=1M or 2M.

I would be glad if some of you could find a better k value.[/QUOTE]

I ran both Sierpinski and Riesel to 1024 with 12-cover and using all primes less than 10 million and no better values were found. I plan to look at 7 or 8 very high k-candidates to see if they can be bettered

 gd_barnes 2008-10-19 08:38

[quote=Siemelink;138059]The riesel conjecture for base 48 = 4208, with cover set {7, 13, 37, 61}
Even => 7
6m+1 => 13
6m+3 => 37
6m+5 => 61

checked n upto 10000
total k 4117
total p 4043
Remaining k 74

I've checked the 4043 primes with pfgw, they all hold up. Of the remaining k, two are squares but I couldn't eliminate them. There is one k that can be divided by 48. but I coudn't eliminate that one either.

Top ten primes
1422 9235
3179 9107
1021 8570
4108 8296
3382 7927
1103 7918
475 7424
2449 7244
3907 7083
3541 7078

All the k's and primes are in the attachment. Feel free to find more primes.

Enjoy, Willem.[/quote]

Per Robert Gerbicz's improved Riesel list, the conjecture for Riesel base 48 is k=3226 with a covering set of {5, 7, 461}. I have now confirmed it.

This reduces the k's remaining at n=10K from 74 to 55 and changed the top 10 primes. The web pages will be updated accordingly.

I checked his list vs. what we show up to Riesel base 50 and that was the only incorrect conjecture that I found.

Gary

[QUOTE=gd_barnes;145817]Per Robert Gerbicz's improved Riesel list, the conjecture for Riesel base 48 is k=3226 with a covering set of {5, 7, 461}. I have now confirmed it.

Gary[/QUOTE]

Oopsie. My own program also gives 3226. I remember I made this riesel by hand and overlooked the smaller possibility. I created my own program because programming is fun and eliminate mistakes like this.

Willem.

 gd_barnes 2008-10-20 21:09

Reserving Riesel base 36. I'll take it up to either n=5K or 10K depending on resource availability.

This is an interesting base because it should be somewhat primeful and is a perfect square, which means that some k's may be eliminated by previous base 6 primes.

Gary

[QUOTE=gd_barnes;145963]Reserving Riesel base 36. I'll take it up to either n=5K or 10K depending on resource availability.

This is an interesting base because it should be somewhat primeful and is a perfect square, which means that some k's may be eliminated by previous base 6 primes.

Gary[/QUOTE]

Hi Gary,

I am also running Riesel 36, I've gotten it to 22500 by now. I think I have said sometime that I was working on it, but I can't find the post just now.
Anyway, as this base is complicated I'd be happy to figure as double check.

I don't have this base quite ready, so I'll post some of it:
Conjecture 116364
Odd 37
6m+2 97
6m+4 43
6m+6 13

1 Attachment(s)
I don't have this base quite ready, so I'll post some of it:
Conjecture 116364
Odd 37
6m+2 97
6m+4 43
6m+6 13[/QUOTE]

And here are the remaining k.

Cheers, Willem.

 gd_barnes 2008-10-22 06:38

I am also running Riesel 36, I've gotten it to 22500 by now. I think I have said sometime that I was working on it, but I can't find the post just now.
Anyway, as this base is complicated I'd be happy to figure as double check.

I don't have this base quite ready, so I'll post some of it:
Conjecture 116364
Odd 37
6m+2 97
6m+4 43
6m+6 13[/quote]

Argh! I'm nearing n=5K and was sieving to n=25K. No, I don't ever remember you stating it or I would have shown it reserved on the web page. I'll stop my effort at n=5K. Of course k==(1 mod 5) and (1 mod 7) as well as k's that are perfect squares are removed.

It's a very nice base for being so large with such a large conjecture. I'm estimating ~60-70 k's will remain at n=100K, although it's a huge effort just to get it that high.

The reason that I was running it is that I have all the base 6 primes and that helped eliminate quite a few k's after I ran it solely on PFGW up to n=2500. But if you've already searched to n=22.5K, that's n=45K base 6, so you've found all but likely the largest 2-3 base 6 primes that apply to base 36...and those are shown on the web page.

Can you please send the primes to me on Riesel base 35 now? I've run it up to n=2K for my usual double-check so if you want to send them all from n=2K to wherever you stopped, then that will be fine. I can't balance it otherwise.

Thanks,
Gary

 gd_barnes 2008-10-22 09:26

Per an Email and 2 PM's here are 138 primes on Riesel base 35 from Karsten for n=5000-6315:

[code]
227548 5007
246796 5007
237026 5026
207388 5035
266522 5044
150560 5054
178946 5060
222458 5060
224968 5061
71858 5068
212746 5069
192416 5070
169846 5075
237196 5075
260926 5085
154166 5094
49184 5100
267614 5116
145588 5117
36086 5120
127228 5121
139430 5122
108124 5129
271048 5131
108466 5143
192284 5144
174560 5152
259324 5169
115748 5182
193802 5192
40364 5198
244702 5205
139136 5210
224942 5214
212150 5232
250916 5238
24274 5243
37456 5247
43610 5264
182840 5270
195176 5274
14974 5277
148646 5278
97582 5301
119960 5302
87948 5303
115324 5311
163582 5313
194582 5326
199396 5331
209464 5341
286294 5341
165668 5368
40096 5371
210322 5373
213482 5382
192818 5394
278260 5397
51314 5402
72016 5413
106616 5420
11508 5421
171614 5426
11570 5430
30202 5445
198350 5476
62078 5484
285658 5511
159958 5519
9194 5540
99698 5564
257884 5601
139562 5602
79498 5607
39692 5622
91778 5640
277520 5664
37898 5676
65534 5680
3628 5683
92768 5690
275070 5712
170572 5727
97098 5734
253340 5734
11672 5736
116660 5752
192890 5758
128948 5776
245150 5794
152462 5798
256688 5804
13936 5819
254998 5819
96142 5823
226164 5823
137062 5825
210976 5837
95600 5864
79004 5894
223232 5898
71578 5899
30196 5933
257062 5945
193774 5947
200710 5957
250060 5963
230948 5964
215972 5966
157576 5969
130472 5986
100348 5987
109810 5995
270332 6000
139814 6002
285308 6006
52282 6009
51972 6010
280268 6016
265658 6032
157420 6033
100294 6035
147878 6044
64808 6066
144920 6076
263630 6102
138518 6108
121222 6125
261524 6150
242362 6173
286394 6176
42992 6256
22920 6259
273698 6264
25924 6279
184408 6291
191776 6301
62252 6306
[/code]

He also tested much higher and found that 94*35^37683-1 is prime.

There are now 1419 k's remaining at n=6315.

Gary

 kar_bon 2008-10-22 09:35

Riesel base 35

next PRPs:

273182 6332
281584 6335
266722 6343
258072 6349
184976 6350
177514 6369
197486 6390
260638 6407
271946 6412
5396 6416
84568 6441

now at n=6462

 gd_barnes 2008-10-22 09:54

1 Attachment(s)

I am also running Riesel 36, I've gotten it to 22500 by now. I think I have said sometime that I was working on it, but I can't find the post just now.
Anyway, as this base is complicated I'd be happy to figure as double check.

I don't have this base quite ready, so I'll post some of it:
Conjecture 116364
Odd 37
6m+2 97
6m+4 43
6m+6 13[/quote]

[quote=Siemelink;146048]And here are the remaining k.

Cheers, Willem.[/quote]

Go ahead and continue with it. My double-check will stop at n=5K plus some other verifications. I'm currently at n=4600. In doing the verification, I found several problems in your list of primes:

k's with lower primes found than in your list (I proved both mine and yours here):
k-value / my prime n= / your prime n=
14503 / 2340 / 6860
102829 / 2276 / 2414
107285 / 3837 / 4121
115402 / 3416 / 3464

typos or incorrect conversions from base 6 primes:
k-value / my prime n= / your prime n= / comment
19389 / 9119 / 9619 / n=9619 has factor 15017; base 6 prime of n= 18238 / 2 = 9119.
19907 / 8439 / 16878 / n=16878 has factor 37; forgot to divide base 6 prime of n=16878 by 2.
94059 / 2352 / 23052 / n=23052 composite, no small factor; extra "0" in n-value.

Here is the only problem that might affect your current testing:
You have 109772*36^11422-1 as prime. It has a factor of 19. But the k-value right above it is also shown with an n=11422 prime; i.e. 109710*36^11422-1. k=109710 is a converted base 6 prime that I ran a primality proof on.

Bottom line: If you removed k=109772 from your testing, you'll need to add it back in and retest starting at n=11422 (I assume).

Also, can you check your machines to see if they may be missing primes or if you just didn't test certain ranges? The prime for k=14503 was particularly troubling because of the n=4520 difference between mine and your primes. If a prime is completely missed, it could result in a huge amount of additional testing. It's not a really big deal if we don't find the smallest prime (although it is my preference) but it makes me wonder about testing when the smallest is not listed.

The final verification I'm doing is running primality proofs on your entire list to make sure there are no additional composites as a result of typos or other things. It's at n=5K right now with no problems found so far.

Adding k=109772 back in leaves 103 k's remaining at n=22.5K for Riesel base 36. Most excellent for a large base and conjecture.

And finally...your list of k's remaining at n=2000 exactly matched mine. I had almost the exact same spreadsheet/document that you did with less primes found due to lower testing limit, including an exact match on all of the converted Riesel base 6 primes. Very good! :smile:

Attached is your list updated with the above corrections.

Gary

 gd_barnes 2008-10-23 01:35

[quote=Siemelink;146048]And here are the remaining k.

Cheers, Willem.[/quote]

Willem,

After making the corrections to your primes noted in the previous post that I made, I ran primality proofs on your entire list. I found one problem:

19315*36^12815-1 is composite

After finishing my testing up to n=5K and finding no prime for the k, I used my own sieved file for n=5K-25K and found a much smaller prime for it:

19315*36^6319-1 is prime

One more small anamoly that isn't really a problem:

I found a smaller prime for k=98420 at n=4722 vs. your n=4965.

Please correct the file that I posted last time to reflect these changes.

One thing you might consider that I do to guarantee finding the smallest prime on each k but also has the added benefit of reducing overall testing time: When testing a base across multiple cores, split the cores up by k-value instead of n-value. For instance, on my testing for this base, I used 2 cores, #1 running k=1 to 58K and #2 running k=58K to 116K. If you split it by n-range, inevitably you end up testing more than you need to.

Also when testing relatively low n-ranges, perhaps n<10K, where many primes are found such as this, I use PFGW even AFTER sieving instead of LLR despite the fact that it is 10-15% slower. The option to make it stop testing a k when it finds a prime means much less manual intervention and mostly offsets the slower testing time. It makes for much cleaner tests. Obviously at the higher n-ranges that take much longer per test where few primes are found, LLR or Prhot are better.

I hope this helps...

Thanks,
Gary

 mdettweiler 2008-10-23 19:34

I'll reserve all 31 remaining k's for Riesel base 37--this looks like an easy base to prove, so I think I'll aim to take it to at least n=20K. :smile:

Edit: I've moved this post to the "Bases 33-100" thread (it was originally in the Reservations/Statuses thread) since it would probably be more appropriate in the former. :smile:

[QUOTE=gd_barnes;146120]Go ahead and continue with it. My double-check will stop at n=5K plus some other verifications. I'm currently at n=4600. In doing the verification, I found several problems in your list of primes:

Gary[/QUOTE]

Yup, that is exactly why I hadn't posted it. I have done some of the doublechecking, but I am no longer clear on what...
The effort is running on a PC with limited access, so no correction possible on the sieve file. I'll wait it out until 25,000 and after that I'll wrap up.

Cheers, Willem.

 robert44444uk 2008-10-24 05:40

Really interesting stuff

1 Attachment(s)
Some conjectures on the lowest Riesel R for base b:

A. Lowest conjectured Riesels are related according to modular arithmetic on the base value.

The graph below which plots (x-axis) b= base, and y-axis lnln(R) with R = lowest conjectured Riesel, suggests modular patterns and I have discovered the following modular relationships:

The following must be taken in order (eg 428==142mod143 but also 32mod33)
Note: ? are conjectured.

R = 4 b==14mod15
R = 6 b==34,69mod105
R = 8 b==20mod21
R = 10 b==32mod33
R= 12 b==142mod143
R = 14 b==8,38,47,64,77,83,116,122,129 or 155mod195
R = 16 b==50,84,152,203,305?,339?,407?,458?mod765
R = 20 b ==18,37,56,75mod95
R = 22 b== 36?,45? and 68mod69
R = 24 b==114mod115
R = 28 b==8? or 86mod87
R = 32 b==30?,61? or 90mod93
R = 34 b== 20?, 21mod33
R = 36 or 38 b==36,73 or 110 mod111
R = 40 when b==122mod123
etc

The case of R = 36 and 38 is curious.

Odd k relationships seem to be less easy to spot, but that is possibly because of the low number of candidates. But I would conjecture:

R = 13 when b==20,38mod264
R = 21 when b==54mod110

This seems to suggest a much easier way to get to low Riesels!! All that is needed is to check the above, or better still, generate the modular algorithm and run through a modular sieve.

B. I wonder is there is a max lnln value for a lowest Riesel? Possibly not, I would guess as I have encountered some nasty looking b's

C. There is an intriguing hole in the graph, tending to lnln(b)=2

Regards

Robert Smith

 robert44444uk 2008-10-24 07:44

Now I understand a bit more about what I am doing, I looked at the Sierpinski side, and found the following modular relationships, which relate to the multiple of the primes in the cover set of the lowest conjectured Sierpinski for a given b. There is at least one anomaly with b=64.

k b
4 14mod15
6 34mod35
8 20mod21: 47,83mod195: 77mod73815: 137mod1551615
10 32mod33
12 142mod143: 562,828,900mod1729: 563mod250705: 597mod1885
901mod19019
13 132,293mod595
14 38mod39: 64mod65 (but not b=64 where k=51!!!)
16 50mod51: 84mod85: 38,47,98,242mod255
18 322mod323: 512mod263683
20 56mod57: 132mod133
21 54mod55
22 68mod69: 160mod161
23 182,878mod795
24 114mod115
25 38mod39
27 90mod91: 538mod2555
28 86mod87
30 898mod899
32 92mod93: 483,747mod2255: 542mod615: 340mod341
34 54mod55: 76mod77
36 159mod233285: 184mod185: 258mod259: 783,993mod1295
38 110mod111: 480mod481: 948mod1105
40 122mod123: 532mod533: 788mod1599

I will go back and redo the Riesel side

 robert44444uk 2008-10-24 08:34

Here is the correct list for Riesels:

k b
4 14mod15
6 34mod35
8 20mod21: 83,307mod455:
10 32mod33
12 142mod143: 901mod19019
13 38,47mod255
14 38mod39: 64mod65: 8,47,83,122mod195
16 50mod51: 84mod85
18 3322mod323: 577mod1105
20 56mod57: 132mod133
21 54mod55
22 68mod69: 160mod161: 657mod3395
24 114mod115
27 90mod91: 922mod4745
28 86mod87: 443mod2355
29 908mod91455
30 898mod899
32 92mod93: 340mod341
34 54mod55: 76mod77: 746mod3471
35 50mod51
36 184mod185: 258mod259
38 110mod111: 480mod481:
40 122mod123

It looks very similar to the Sierpinski list

 gd_barnes 2008-10-24 09:20

[quote=Siemelink;146313]Yup, that is exactly why I hadn't posted it. I have done some of the doublechecking, but I am no longer clear on what...
The effort is running on a PC with limited access, so no correction possible on the sieve file. I'll wait it out until 25,000 and after that I'll wrap up.

Cheers, Willem.[/quote]

With only one exception, all the composites that I found on your list turned out to be typos with other primes. Therefore, I tested the one exception, k=109772, up to n=25K. No prime was found. That was the k that you had repeated the n-value prime from the k-value right about it in your list.

I also continued my double-check up to n=6.2K and found no additional problems.

So double-checking to n=6.2K and running primality proofs on your entire list confirms that there are definitely 103 k's remaining for Riesel base 36 at n=22.5K after subtracting off the converted base 6 higher primes that you have already done.

Gary

 gd_barnes 2008-10-24 09:39

[quote=mdettweiler;146310]I'll reserve all 31 remaining k's for Riesel base 37--this looks like an easy base to prove, so I think I'll aim to take it to at least n=20K. :smile:

Edit: I've moved this post to the "Bases > 32 that are not powers of 2" thread (it was originally in the Reservations/Statuses thread) since it would probably be more appropriate in the former. :smile:[/quote]

I've had this type of discussion with others, especially Kenneth. People well under-estimate the difficulty of finding primes for low weight k's on high bases, especially at high n-values.

Example: I'm testing the ONE final k on Sierp base 12 at n=195K. I'm going to n=250K on a file already fully sieved to P=7T (which took quite a while to get sieved that high), with likely less than a 20% chance of prime by that point. Testing time per candidate on one of my highest-speed machines: Nearly 1 hour! Total CPU time needed to complete it: 80-85 days and that's for just one k on a base 1/3rd as high as yours!! Running on just one core, it is crawling along. Sometime when it passes n=200K; I'll probably put 4 quads on it and knock it out in ~4-5 days just to get it off my plate but that's a LOT of firepower on one simple k for only an n=50K range!

Also, base 37 is not base 31 or base 36. It's not a very prime base at all. Example: There were 41 k's remaining at n=3.2K and 31 k's remaining at n=10K. If you assume a 25% reduction in k's remaining for every tripling of the n-value, you have:

n=10K; 31 k's remain
n=30K; 23 remain
n=90K; 17 remain
n=270K; 13 remain

To get this base to n=100K will be a huge, although very worthwhile effort. As for proving it, even if I'm way off and you only have 4-5 k's remaining at n=270K; this is likely > 100 CPU-year effort to prove it. All that I can say is: Good luck! You'll need it. :smile:

Gary

 gd_barnes 2008-10-24 09:47

Primes and status in a PM from Karsten on Riesel base 35:

[quote]
next ones:
269516 6470
81250 6473
65874 6486
166052 6492
32954 6498
163094 6504
236596 6513
231208 6521
272612 6526
281816 6536
232064 6538
227888 6556
48244 6565
110566 6565
225482 6576
148766 6588
128626 6593
166208 6596

at n=6606 with 5.19M candidates left (sieved to 4.3G, will sieving further)
[/quote]

Gary

 kar_bon 2008-10-24 10:48

[QUOTE=robert44444uk;146357]
(...)
16 50mod51: 84mod85
18 3322mod323: 577mod1105
20 56mod57: 132mod133
(...)
[/QUOTE]

perhaps a misprint?!

should be: 18 [b]322[/b]mod323: 577mod1105

 kar_bon 2008-10-24 10:56

Riesel Base 35

next PRP-values:

161266 6625
271540 6631
209114 6652
225590 6656
240320 6684
14204 6714
57170 6720
155966 6720
99862 6733

at n=6735

 robert44444uk 2008-10-24 13:10

[QUOTE=kar_bon;146364]perhaps a misprint?!

should be: 18 [b]322[/b]mod323: 577mod1105[/QUOTE]

Certainly a misprint from my Excel spreadsheet. There maybe others, hopefully not.

Also I notice that 54mod55 turns up twice, (k=21 and 34) - the second value comes into play if k=21 provides a facile result, both are related to covers [5,11]. And that I now know also explains the base 64 problem.

So the mods I show provide a theoretical low k value and if that produces a facile result then you have to look at other mod combinations. With these low k you don't have to look far.

Dan Krywaruczenko will soon publish his work on determining any k as a Sierpinski/Reisel value excluding k=Mersenne.

 mdettweiler 2008-10-24 14:22

I've had this type of discussion with others, especially Kenneth. People well under-estimate the difficulty of finding primes for low weight k's on high bases, especially at high n-values.

Example: I'm testing the ONE final k on Sierp base 12 at n=195K. I'm going to n=250K on a file already fully sieved to P=7T (which took quite a while to get sieved that high), with likely less than a 20% chance of prime by that point. Testing time per candidate on one of my highest-speed machines: Nearly 1 hour! Total CPU time needed to complete it: 80-85 days and that's for just one k on a base 1/3rd as high as yours!! Running on just one core, it is crawling along. Sometime when it passes n=200K; I'll probably put 4 quads on it and knock it out in ~4-5 days just to get it off my plate but that's a LOT of firepower on one simple k for only an n=50K range!

Also, base 37 is not base 31 or base 36. It's not a very prime base at all. Example: There were 41 k's remaining at n=3.2K and 31 k's remaining at n=10K. If you assume a 25% reduction in k's remaining for every tripling of the n-value, you have:

n=10K; 31 k's remain
n=30K; 23 remain
n=90K; 17 remain
n=270K; 13 remain

To get this base to n=100K will be a huge, although very worthwhile effort. As for proving it, even if I'm way off and you only have 4-5 k's remaining at n=270K; this is likely > 100 CPU-year effort to prove it. All that I can say is: Good luck! You'll need it. :smile:

Gary[/quote]
Oh, I see. :sad: I had assumed that with only 31 k's remaining at n=10K (which, for a lower base, would have meant quite good chances of proving it quickly, even without the advantage of a very prime base), my odds were pretty good--I guess not. Oh well--I'll still take it up to n=20K and see what I can knock out. :smile:

 gd_barnes 2008-10-25 10:56

[quote=mdettweiler;146384]Oh, I see. :sad: I had assumed that with only 31 k's remaining at n=10K (which, for a lower base, would have meant quite good chances of proving it quickly, even without the advantage of a very prime base), my odds were pretty good--I guess not. Oh well--I'll still take it up to n=20K and see what I can knock out. :smile:[/quote]

What difference would it make if it was a very low base; even base 2 or 3? Virtually none at all.

Obviously I still haven't made myself clear on the difficulty in proving these things. Please take a look at the most prime base of all: base 3. On the Sierp side, we'll likely knock out almost exactly half of k's remaining testing k<50M from n=35K to n=100K so for the purposes of estimate, we'll just assume that we halve k's remaining for every 3X increase in n-range; therefore if 31 k's were remaining at n=10K:

10K; 31 k's remaining
30K; 16 k's remaining
90K; 8 k's remaining
270K; 4 k's remaining
810K; 2 k's remaining
1.62M; 1 k remaining
3.24M; 0.5 k remaining

So likely, you'd have to test it to n=3M. Doable as a project over several years but not as an individual.

The point is that 31 k's is a huge # of k's remaining in ANY base at n=10K! For any one person to have a shot at proving a base, there needs to be < ~10 k's remaining at n=10K.

The reason that I want to make this so clear is that people have a tentency to reserve far more than they will ever want to complete. Your reservation to n=20K is quite reasonable for total workload but gives no chance of proving the conjecture in any base for so many remaining k's at n=10K.

Karsten, are you still going to take Riesel base 6 to n=1M? (lol)

Gary

 mdettweiler 2008-10-25 17:11

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. :smile:

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. :smile:

 KEP 2008-10-25 20:04

@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 :smile:

KEP

 henryzz 2008-10-25 20:33

your post has helped massively thanks gary

 mdettweiler 2008-10-25 20:38

[quote=mdettweiler;146478]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. :smile:

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. :smile:[/quote]
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 :w00t:

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

 gd_barnes 2008-10-26 06:56

[quote=mdettweiler;146513]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 :w00t:

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

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 2008-10-26 13:22

I get it now. :smile: 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. :smile:

BTW, found another one last night:

498*37^15332-1 is prime!

 gd_barnes 2008-10-26 20:19

[quote=mdettweiler;146625]I get it now. :smile: 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. :smile:

BTW, found another one last night:

498*37^15332-1 is prime![/quote]

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. :smile:

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 2008-10-26 21:00

[quote=gd_barnes;146656]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. :smile:

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[/quote]
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 2008-10-27 10:02

Riesel Base 35

new PRPs over weekend:
[code]
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
[/code]

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

 kar_bon 2008-10-27 12:36

[QUOTE=gd_barnes;146592]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.
[/QUOTE]

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 2008-10-28 04:17

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

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

 gd_barnes 2008-10-28 23:59

[quote=kar_bon;146770]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![/quote]

Very interesting info. Karsten. We should be able to do an accurate analysis of future k's remaining based on this info.

Gary

 kar_bon 2008-11-03 09:53

Riesel Base 35

new PRP's
[code]
98114 7140
186752 7160
26522 7162
193960 7171
141602 7180
187898 7216
170470 7219
81038 7222
141144 7239
154090 7261
229660 7263
197416 7267
125242 7269
216830 7302
204914 7342
65864 7346
86624 7366
215398 7379
28010 7382
167314 7387
197042 7390
97942 7391[/code]

4.89M pairs left upto n=7420

 kar_bon 2008-11-05 08:41

Riesel Base 35

new PRPs
30304 7423
233318 7426
239534 7438
240080 7450
7478 7452

n=7454

 kar_bon 2008-11-17 11:44

Riesel Base 35

267464 7464
95720 7478
63878 7482
82414 7497
53192 7542
25888 7545
11738 7558
162698 7566
230324 7572
72454 7591
258004 7609
111230 7638
259240 7657
53290 7659
9716 7684
122840 7698
209296 7713
93154 7723
198856 7733
163276 7749

at n=7765

 kar_bon 2008-11-27 09:32

Riesel base 35

new PRP's

205298 7772
188126 7776
184432 7813
156122 7830
62312 7856
70648 7875
190930 7879
96422 7882
63388 7887
205214 7896
180190 7899
75238 7903
10808 7912
132392 7926
134092 7937
140084 7966
113006 7984
46874 7996
134240 8010

now at n=8024 with 4.675M candidates left

 henryzz 2008-11-27 09:49

does anyone have any suggestions on what bases it would be easiest to prove

 MrOzzy 2008-11-27 10:56

You don't know if a conjecture will easely be proven before you actually start to prove it (just look at sierp base 17 and 18 for example).
I can give you a list of bases with a relatively low conjectured k (<1000).
The first number is the base and the number between brackets is the conjectured k.

Conjectured k for bases 51 to 100:

Sierp: 54 (21), 56 (20), 59 (4), 62 (8), 64 (51), 65 (10), 68 (22), 69 (6), 72 (731), 74 (4), 76 (43), 77 (14), 83 (8), 84 (16), 86 (28), 89 (4), 90 (27), 92 (32), 94 (39), 98 (10), 99 (684)

Riesel: 54 (21), 56 (20), 57 (144), 59 (4), 62 (8), 64 (14), 65 (10), 68 (22), 69 (6), 72 (293), 73 (408), 74 (4), 77 (14), 80 (253), 81 (74), 83 (8), 84 (16), 86 (28), 89 (4), 90 (27), 92 (32), 93 (612), 94 (39), 98 (10), 99 (144), 100 (750)

You can also for example go for Briers ([URL]http://www.mersenneforum.org/showthread.php?t=10930[/URL]) or try to prove the first, second, third, ... conjectured k for one specific base with a lot of small conjectured k (Riesel base 68 for example has a conjectured k at k=22, 43, 142, 185, 783, 1394, 3051)

If you need more info, just ask. I have a lot more interesting things you can do with conjectures :)

 gd_barnes 2008-11-27 17:17

I've recently been working on a # of the easier unreserved Riesel bases 50 thru 125. The Sierp side is open for bases > 50 although we have some info. already from Prof. Caldwell for bases 50-100.

I'm going to post the results of some of my searches later tonight. Some were very easily proven and a few others have just a few k's left and could be proven by others at some point.

There is a thread that has all of the conjectured values for all bases on both sides up to 1024. That would be a good starting point.

Gary

 gd_barnes 2008-11-27 17:20

[quote=MrOzzy;150940] prove the first, second, third, ... conjectured k for one specific base with a lot of small conjectured k (Riesel base 68 for example has a conjectured k at k=22, 43, 142, 185, 783, 1394, 3051)
[/quote]

This is a very interesting idea that I have toyed around with at different times but never stuck with it very long. Riesel and Sierp base 8 would be interesting bases to attack to prove the 2nd/3rd/etc. conjectured k's since their 1st one is so low and was already easily proven. Also, since base 8 is a power of 2, LLRing would be fast.

Gary

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