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Bases 33-100 reservations/statuses/primes
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Admin. edit: We have started this new thread for bases 33-100. See more info. in [URL="http://www.mersenneforum.org/showpost.php?p=137717&postcount=14"]post 11[/URL] of the effort here.
[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. |
bases 33-100
[quote=Siemelink;137058]I have already finished this one.
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
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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 |
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. Please post all primes, statuses, and results in this thread. When reporting things, please account for all k-values in some manner. This will help keep the admin. effort to a minimum. :smile: Gary |
[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. |
[quote=Siemelink;137738]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.[/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
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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. |
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 |
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
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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. |
[quote=Siemelink;138248]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.[/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
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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. |
[quote=Siemelink;138726]
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
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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. |
[quote=Siemelink;139767]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. 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
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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
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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. |
[quote=Siemelink;144576]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.[/quote] Can you please send the primes to me at: gbarnes017 at gmail dot com Thanks, Gary |
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. |
reserved base 35 from n=5k-100k
so Siemelink can check another base :-) |
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=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 |
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. |
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=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! |
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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=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] Good work Siemelink :smile: 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 |
[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 :-) |
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! |
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 |
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. |
[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 |
[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 |
[QUOTE=R. Gerbicz;145585]Better Riesel values, also up to base=1024:
[URL="http://robert.gerbicz.googlepages.com/riesel.txt"]http://robert.gerbicz.googlepages.com/riesel.txt[/URL][/QUOTE] 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. |
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] |
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 |
[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. |
[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. |
[quote=Siemelink;144576]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.[/quote] Willem, Can you send me all your primes on Riesel base 35? Thanks, Gary |
[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 |
[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 |
[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. |
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 |
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[QUOTE=Siemelink;146047]
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. |
[quote=Siemelink;146047]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[/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 |
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 |
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 |
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[quote=Siemelink;146047]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[/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 |
[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 |
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. |
Really interesting stuff
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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 |
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 |
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 |
[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 |
[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] Not likely to be proven in your lifetime! (not joking) 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 |
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 |
[QUOTE=robert44444uk;146357]
(...) 16 50mod51: 84mod85 18 3322mod323: 577mod1105 20 56mod57: 132mod133 (...) [/QUOTE] perhaps a misprint?! should be: 18 [b]322[/b]mod323: 577mod1105 |
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 |
[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. |
[quote=gd_barnes;146361]Not likely to be proven in your lifetime! (not joking)
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: |
[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 |
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: |
@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 |
i too have been thinking wrong about this sort of thing
your post has helped massively thanks gary |
[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: |
[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 |
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=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 |
[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. |
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 |
[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! |
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. |
[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 |
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 |
Riesel Base 35
new PRPs
30304 7423 233318 7426 239534 7438 240080 7450 7478 7452 n=7454 |
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 |
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 |
does anyone have any suggestions on what bases it would be easiest to prove
|
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 :) |
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 |
[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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