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 2005-04-05, 21:53 #1 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts Octoproths I am posting this again for those looking for a challenge - but this time in the Maths thread: Try to find the smallest and largest integer (maybe larger than you think) k*2^n+1 such that: k*2^n+1, k*2^n-1, k*2^(n+1)+1, k*2^(n+1)-1, 2^n+k, 2^n-k, 2^(n+1)+k, 2^(n+1)-k, all probable prime I have called these octoproths. On the large side of things, I have looked at n=32 and 1
 2005-04-06, 22:58 #2 robert44444uk     Jun 2003 Oxford, UK 2·7·139 Posts A bit bigger Maybe I am just being a bit shy, but I did not raise n too much but I devoted a whole 8 hours to sieving and came across a lot of larger values at n=40, namely: 3721055715 8781593205 9073509705 15541397325 20640857145 23820338205 27678219015 28483380255 29056766445 34144356345 38016547275 38354659875 43635963015 43838018925 48817120275 57779210775 58422340185 58443226395 59679412785 61308889305 62456594325 64989748725 76695589755 76715178345 79069804605 89329921695 92992163025 93765951525 97257415185 98114137575 99030787695 They are not all divisible by 105, but are all divisible by 15 I think this means that tonight my compuiter will sieve at the n=50 level Regards Robert Smith
 2005-04-07, 08:40 #3 axn     Jun 2003 5,081 Posts k=925905105, n=64 925905105*2^64+1 925905105*2^64-1 925905105*2^(64+1)+1 925905105*2^(64+1)-1 2^64 + 925905105 2^64 - 925905105 2^(64+1) + 925905105 2^(64+1) - 925905105 All are prime!! Searched for 1 < k < 2^32
 2005-04-07, 10:31 #4 axn     Jun 2003 5,081 Posts k=3539387145, n=65
 2005-04-07, 10:47 #5 Templus     Jun 2004 10610 Posts How do you find these 'k'-s? Do you first sieve for a range 1 < k < 2^32 with a given 'n' for the form k*2^n+1, and after that you use that sieve as seed for a sieve for the form k*2^n-1? And use those results as seed for another sieve for another form, etcetera etcetera? Or is there are another (sieve-)program that can help? Quite interesting numbers though! Last fiddled with by Templus on 2005-04-07 at 10:47
 2005-04-07, 12:04 #6 axn     Jun 2003 5,081 Posts I used NewPGen BiTwin option which sieves for the 4 forms: k.2^n+/-1 and k.2^(n+1)+/-1 This reduces the billions of k's to about 10000. It is then run thru LLR, first for k.2^n+1, whose output is again LLR-ed for k.2^n-1, whose output is *again* LLR-ed for k.2^(n+1)+1, and finally (you guessed it!) LLR-ed for k.2^(n+1)-1 The resulting k's are then normalized (I sieve for even k's also) to odd k's. Then after a bit of script magic and Dario Alpern's batch factorization, I reduce the list to 2^n+k primes, then to 2^n-k primes, and finally the last two forms to come up with the solutions. Incidentally, just completed n=66 and no solutions Oh well. On to 67 !! Last fiddled with by axn on 2005-04-07 at 12:16
 2005-04-07, 13:00 #7 axn     Jun 2003 5,081 Posts Nothing for n=67 either
 2005-04-07, 16:08 #8 robert44444uk     Jun 2003 Oxford, UK 2·7·139 Posts Another way I also use the bitwin option in NewPGen to reduce to an acceptable number of candidates. Sieving to 1 billion is fine, which is the minimum p which the large k range option NewPGen uses. My NewPgen splits the range into 120 or so smaller ranges and then batches all of the survivors together when each range is tested to 1 billion. At the next stage, I do not check the output for primes. Instead I alter the first line of the NewPgen output file to make this into an ABC file, and use the & option to check for the four other values of the octoproth. This file is then put through pfgw, with the -f100 option, which, for n=50 checks around 10,000 values of k a second. The output file for this run can be inspected to see if there are any values where all four possibilities are prp. It is quicker this way because the pfgw will give up testing the value of k as soon as it spots a composite. If you test the original output file then a lot of values will have to be tested for all four options as none of them, if they are composite, have factors of less than 1 billion. The final stage is to extract the few (maybe only ten values) where the output file shows the complete set is prp, and make this into a PFGW input file with the first line of the file the same as the output file from the bitwin output file. About half of the values will be octoproths. Therefore almost all of the work is in the sieving because a very large number of k must be looked at, for a given n. My computer takes 8 hours to sieve k=2 to k=10^11 up to p=1 billion. A further 2 hours takes the file to p=30 billion. I ran this range for n=50, and found about 8 octoproths. But it looks as if I have been overtaken by events. The bar has been raised! Regards Robert Smith
 2005-04-07, 18:01 #9 Templus     Jun 2004 2·53 Posts I tried this method for n=81 and 2 < k < 2^32. I made a sieve file which contains about 16000 lines, but I'm not sure about the ABC rule I have to construct. I now have: Code: ABC $a*2^$b+1 & $a*2^$b-1 & $a*2^($b+1)+1 & $a*2^($b+1)-1 & 2^$b+$a & 2^$b-$a & 2^($b+1)+$a & 2^($b+1)-$a Is this definition correct? When I run this code and I check the pfgw.log file I only see lines of the form k*2^n+1, k*2^n-1, k*2^(n+1)+1, k*2^(n+1)-1 but not of the form 2^n+k, 2^n-k, 2^(n+1)+k, 2^(n+1)-k. What am I doing wrong?
 2005-04-07, 21:35 #10 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts Answer The line I use is (followed by first putput from the NewPGen file): ABC 2^$b+$a & 2^$b-$a & 2^($b+1)+$a & 2^($b+1)-$a 708435 40 Regards Robert Smith
2005-04-08, 03:48   #11
axn

Jun 2003

5,081 Posts

Quote:
 Originally Posted by robert44444uk My computer takes 8 hours to sieve k=2 to k=10^11 up to p=1 billion. A further 2 hours takes the file to p=30 billion.
How do you sieve with k as high as 10^11. I thought that NewPGen only supported upto k < 2^32 !

 2005-04-08, 07:23 #12 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts Dont know I didn't know there was an upper limit, I just do it! I am using version 2.81, and I set max Mb ram to 100.0 I ran n=71 last night - only one value survived the sieve and the first pfgw run, and that turned out to be composite for k.2^71-1 So Anx1's record still stands. Regards Robert
 2005-04-08, 08:53 #13 Templus     Jun 2004 2×53 Posts I tried n=81 yesterday for 2
 2005-04-08, 10:49 #14 ltd     Apr 2003 14048 Posts Are n=74,75,76 free ? I would like to test them. Lars
2005-04-08, 11:00   #15
axn

Jun 2003

5,081 Posts

Quote:
 Originally Posted by robert44444uk I didn't know there was an upper limit, I just do it! I am using version 2.81, and I set max Mb ram to 100.0
Hmmm.... Must be the version. I am using 2.82. If I put k higher than 2^32-1 it just wont budge :

But on the plus side, I went and wrote up a sieve program for myself. The initial results are *very* encouraging. I sieve all 8 forms simultaneously.

Some statistics. After sieveing for k < 10^10, with p < 1M, I get just 90 (count'em) candidates

 2005-04-08, 11:35 #16 axn     Jun 2003 508110 Posts And the winner is: k=61574201535, n=80
2005-04-08, 13:26   #17
robert44444uk

Jun 2003
Oxford, UK

79A16 Posts
For Lars

Quote:
 Originally Posted by ltd Are n=74,75,76 free ? I would like to test them. Lars
You might want to try a bit higher, given the latest record. This is addictive!!

Regards

Robert Smith

 2005-04-08, 15:17 #18 ltd     Apr 2003 14048 Posts OK i take 100,101,102 Lars
 2005-04-08, 16:01 #19 axn     Jun 2003 5,081 Posts Another one for n = 80 k = 632893190475 !!
 2005-04-08, 16:17 #20 axn     Jun 2003 5,081 Posts n = 80 finished for k < 10^12. On to bigger n's
2005-04-08, 21:26   #21
ltd

Apr 2003

22·193 Posts

Quote:
 Originally Posted by axn1 Hmmm.... Must be the version. I am using 2.82. If I put k higher than 2^32-1 it just wont budge :
Strange i use 2.82 also and i have no problems sieving >2^32.

2005-04-09, 06:21   #22
axn

Jun 2003

5,081 Posts

Quote:
 Originally Posted by ltd Strange i use 2.82 also and i have no problems sieving >2^32.
D'oh! It looks like I've run into a bug in NewPGen. Try putting kmax = 4294967296 (2^32). It doesnt do anything. If you put any other number in there (including really big numbers), it springs into action!

Well, anyway, my new sieve is a lot better suited for this search.

 2005-04-09, 10:31 #23 axn     Jun 2003 5,081 Posts For n=82, there are 4 k's < 10^12 42290329515 481562533725 549711786105 624949113615 EDIT: No luck for n=81 for k < 10^12. This n was a "low weight" one compared to n=82 Last fiddled with by axn on 2005-04-09 at 10:41
 2005-04-09, 13:59 #24 ltd     Apr 2003 22×193 Posts No luck for n=100,101,102. @axn1 What OS are you using for your siever programm. If it is windows is it possible that i can download it somewhere? Lars
 2005-04-09, 14:34 #25 robert44444uk     Jun 2003 Oxford, UK 2·7·139 Posts no luck either I did n=110 last night to k=10^11, no octoproths thier either Regards Robert Smith
 2005-04-10, 08:46 #26 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts 109 Checked 109 last night, one candidate that fell at the last hurdle, unlike the horse I chose for the Grand National, which cost me a tenner when it fell at the first! Will do 108 tonight. Regards Robert Smith
 2005-04-11, 02:51 #27 TTn   2×3×557 Posts interesting Robert, Nice find, I will have to check it out. I have been touching up RMA a bit, and have'nt been online to catch up on what's going on. TTn
2005-04-11, 09:04   #28
axn

Jun 2003

13D916 Posts

Results for n = 97 (after 65% completion to k<10^13)

1926973493115
2212009461375
2412877121565
5647136892825

@ltd - see the attached Pascal source code - Its not much. You'll have to modify the constants in the program and compile (you can use FreePascal compiler).

I plan to later clean it up and make it accept command line parameters
Attached Files
 octo.pas.txt (2.3 KB, 230 views)

Last fiddled with by axn on 2005-04-11 at 09:05

2005-04-11, 09:43   #29
ET_
Banned

"Luigi"
Aug 2002
Team Italia

10010110100012 Posts

Quote:
 Originally Posted by axn1 Results for n = 97 (after 65% completion to k<10^13) 1926973493115 2212009461375 2412877121565 5647136892825 @ltd - see the attached Pascal source code - Its not much. You'll have to modify the constants in the program and compile (you can use FreePascal compiler). I plan to later clean it up and make it accept command line parameters
May I ask you which boost of performance gave the substitution of Pascal code with asm code?

Luigi

2005-04-11, 10:11   #30
axn

Jun 2003

5,081 Posts

Quote:
 Originally Posted by ET_ May I ask you which boost of performance gave the substitution of Pascal code with asm code? Luigi
The two divisions in TestK - gave appr. 35% speedup. Not much but I'll take any speedup especially since I am running these for 2-3 days at a stretch. Actually, there is one more optimization there - the division of k by p is done by two back-to-back divisions; for most cases you only need one division. I plan to code it up and try it out. Let's see what kind of improvement it brings. For people needing non-asm version, you can use suitable qword operations. But in such cases, it might be worthwhile to use alternatives to division.

2005-04-11, 11:38   #31
axn

Jun 2003

13D916 Posts

Quote:
 Originally Posted by axn1 Results for n = 97 (after 65% completion to k<10^13) 1926973493115 2212009461375 2412877121565 5647136892825
One more to the list for n = 97

6832047128535

2005-04-11, 13:01   #32
axn

Jun 2003

508110 Posts

Uploading the latest version of the sieve along with the executable. The output needs to be redirected to some file. The resulting file can be further sieved using NewPGen.
Attached Files
 octo.zip (13.2 KB, 227 views)

 2005-04-11, 13:09 #33 robert44444uk     Jun 2003 Oxford, UK 111100110102 Posts 108 Ran 108 last night to 10^11, and no octos, sadly to say. Axn1, will you be writing your code in a windows executable? I would certainly be interesting in devoting some raw computer power to take it further. Regards Robert Smith
 2005-04-12, 03:13 #34 Dougy     Aug 2004 Melbourne, Australia 9816 Posts Here's two small ones I found using axn1's program. 8299358445 50 3920165865 54
 2005-04-12, 03:23 #35 Dougy     Aug 2004 Melbourne, Australia 23·19 Posts And of course as soon as I posted those I found some more... 13419352155 52 14002823745 52 19306888875 52 26648959155 52
 2005-04-12, 03:24 #36 axn     Jun 2003 10011110110012 Posts k = 405777203685, n = 120 Thats the only one for k < 10^13. Robert, the last attachment had a windows executable (console mode). You can run it from cmd prompt and redirect the output to a file.
2005-04-12, 09:31   #37
axn

Jun 2003

13D916 Posts

Quote:
 Originally Posted by axn1 One more to the list for n = 97 6832047128535
The rest for n = 97:

8246997577755
8883883726185
9417272582445
9910177359165

These are the last for k < 10^13

 2005-04-12, 14:58 #38 robert44444uk     Jun 2003 Oxford, UK 2·7·139 Posts Oops Oops should have tried first! However, how do you write the line script to create an output file? It is years since I saw dos. I tried this c:\octo 50 10 and got an output on my screen with about 10 candidates. Are these candidates or are they in fact octos? Sorry to be a bit naive, but I cannot read music and I cannot read other people's computer programs! Regards Robert Smith
 2005-04-12, 15:21 #39 robert44444uk     Jun 2003 Oxford, UK 79A16 Posts odd statistics I ran Axn1's prgram, up to 10^10 for n=50 through 58. The number of candidates (octos?) produced by the programme are: 50 11 51 5 52 47 53 7 54 28 55 27 56 5 57 18 58 17 I wonder what is so special about 52, it seems statistically well outside of normal variances? Regards Robert Smith
 2005-04-12, 18:49 #40 TTn   13×769 Posts Octo RMA1.74 This sounds like a nice easy addition to RMA 1.74, and will be listed under "Preferences" "Other options" "Octoproth". I'll need about a week to get on it. If there are any additional behaviours or options, that you think should be included under the octoproth option, please post them. TTn
2005-04-13, 09:30   #41
axn

Jun 2003

10011110110012 Posts

Quote:
 Originally Posted by robert44444uk However, how do you write the line script to create an output file? It is years since I saw dos.
octo 50 10 > candidates.txt

Quote:
 Originally Posted by robert44444uk I wonder what is so special about 52, it seems statistically well outside of normal variances?
Yes. I too have observed this. A few posts back, I had said that 81 was "low weight" compared to 82. My guess is that for some of the n's, some small prime(s) might be eliminating a lot of candidates. Conversely, some n's might be escaping these small primes.

Incidentally, these "heavy weight" n's all seem to be of the form 3x+1.

 2005-04-14, 07:18 #42 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts New Record! Playing around with Axn1's software has allowed Great Britain to regain the World record for largest octoproth. Hurrah for that, hip, hip, hooray. 374526655755*2^113+1 is 3-PRP! (0.0001s+0.0002s) 374526655755*2^113-1 is 3-PRP! (0.0001s+0.0045s) - Twin - 374526655755*2^(113+1)+1 is 3-PRP! (0.0001s+0.0079s) 374526655755*2^(113+1)-1 is 3-PRP! (0.0001s+0.0045s) - BiTwin - 2^113+374526655755 is 3-PRP! (0.0030s+0.0002s) 2^113-374526655755 is 3-PRP! (0.0001s+0.0067s) 2^(113+1)+374526655755 is 3-PRP! (0.0001s+0.0042s) 2^(113+1)-374526655755 is 3-PRP! (0.0001s+0.0042s) - Complete Set - Regards Robert Smith
 2005-04-14, 07:36 #43 Dougy     Aug 2004 Melbourne, Australia 100110002 Posts Well done robert, they're all prime by the way. However the largest known is k=405777203685 n=120 found by axn1.
 2005-04-14, 08:14 #44 Dougy     Aug 2004 Melbourne, Australia 23×19 Posts Small Octoproths These are the smallest octoproths for their corresponding bases. Why 56 is so large is a real head-scratcher. 8299358445 50 106546113135 51 13419352155 52 216800357445 53 3920165865 54 72038479785 55 590925115935 56 138429315465 57 84183246225 58 107884757295 59
 2005-04-15, 04:26 #45 axn     Jun 2003 5,081 Posts Results for n = 130, (k < 10^13) 1075252753275 3408331609305 7076113724805
 2005-04-15, 08:38 #46 Dougy     Aug 2004 Melbourne, Australia 23×19 Posts Small Octos I've been looking at the small bases. (primes, rather than probable primes) I wrote my own program to look at these. There are no octoproths with base n = 26 or below. The first one is 109989075 27 and is the only one with base 27. The next are 21207165 28 191093475 28 are the only two with base 28. ...more to come One interesting one is n=1, k=15. 15*2^1+1 = 31 15*2^1-1 = 29 15*2^(1+1)+1 = 61 15*2^(1+1)-1 = 59 2^1+15 = 17 2^1-15 = -13 2^(1+1)+15 = 19 2^(1+1)-15 = -11 If you count negative primes too.
 2005-04-15, 15:38 #47 robert44444uk     Jun 2003 Oxford, UK 36328 Posts Really suprised Dougy I am really surprised that there are no "small" octos. The way I have defined them means that negative numbers, created through the 2^n-k calculation, rule that number out, so your interesting case has to remain as that. But thank you for looking at the small case. I just find the result hard to believe, but the negative rule counts out a lot for small n, especially when k goes in multiples of 15 (almost 2^4), so maybe I should have realised. Maybe you should post the full decimal value of this find to Chris Caldwell's Prime curios page: http://primes.utm.edu/curios/ Regards Robert Smith
 2005-04-15, 15:45 #48 robert44444uk     Jun 2003 Oxford, UK 2×7×139 Posts More for Dougy Dougy I just realised, (as I am sure you have) that you will need also to look at higher n, because they may have a smaller k value, such that k.2^n+1 is a smaller number. So that you will have to check almost all the way up to n=50 to make totally sure there are no smaller octos. Regards Robert
 2005-04-16, 01:20 #49 Dougy     Aug 2004 Melbourne, Australia 23×19 Posts Smallest So, if my program works properly, there are no (certified prime) octoproths within the ranges n=31-50 and k=15-21207165. Furthermore 328724235 29 233752995 30 are the only octoproths with those bases. So this is a proof that 21207165*2^28+1 = 5692755007242241. 109989075*2^27+1 = 14762483751321601. are the smallest two octoproths. Also 21207165 is also the smallest known k-value forming a octoproth. I wonder if it's actually the smallest possible. I might search with a fixed k and varying n instead. (but that'd require writing a whole new program) It would be nice if someone could verify this independently before I submit it anywhere.
 2005-04-16, 03:30 #50 TTn   29×127 Posts oct I working on the new version now...

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