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^n1, k*2^(n+1)+1, k*2^(n+1)1, 2^n+k, 2^nk, 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<k<4 billion and found only 6, with the following k values: 409668105 664495755 2368386195 2709707805 3383804865 3692088225 Interestingly, but not surprisingly these k are all multiples of k=105 (3*5*7) as this is a requirement for bitwins (I think). And therefore of interest to 15k searchers. I have no idea about the smallest octoproths. The really interesting thing about these groups is that it combines twins, Cunninghams and Payam Samidoost's observation about 2^n+/ k, these have the same covering sets as their Proth equaivalents. Watch out for negative values, because 2^nk can be really small Regards Robert Smith 
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 
k=925905105, n=64
925905105*2^64+1 925905105*2^641 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 
k=3539387145, n=65

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^n1? 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! 
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 LLRed for k.2^n1, whose output is *again* LLRed for k.2^(n+1)+1, and finally (you guessed it!) LLRed 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^nk primes, and finally the last two forms to come up with the solutions. Incidentally, just completed n=66 and no solutions :no: Oh well. On to 67 !! 
Nothing for n=67 either :sad:

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 
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 & $a*2^$b1 & $a*2^($b+1)+1 & $a*2^($b+1)1 & 2^$b+$a & 2^$b$a & 2^($b+1)+$a & 2^($b+1)$a [/code] 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^n1, k*2^(n+1)+1, k*2^(n+1)1 but not of the form 2^n+k, 2^nk, 2^(n+1)+k, 2^(n+1)k. What am I doing wrong? 
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 
[QUOTE=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.
[/QUOTE] How do you sieve with k as high as 10^11. I thought that NewPGen only supported upto k < 2^32 ! 
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.2^711 So Anx1's record still stands. Regards Robert 
I tried n=81 yesterday for 2<k<2^32. As far as I know, there is no upperbound, since NewPGen splits this large range into smaller pieces (dependent on the max. RAM assigned?) and sieves these ranges until 1 billion.
As I said, I tested n=81 yesterday with OpenPFGW and there was no complete set in the logfile. I still have the sieve file, so anyone who wishes to redo it can ask me for the sievefile. I uploaded the sieve file on [URL]http://www.diamondvalley.nl/octoproth[/URL] I'm taking 82+....sieving until k=2^32 takes about 1,5 or 2 hours for me. Testing these numbers for PRP is done in less than a minute (small n) 
Are n=74,75,76 free ?
I would like to test them. Lars 
[QUOTE=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 [/QUOTE] Hmmm.... Must be the version. I am using 2.82. If I put k higher than 2^321 it just wont budge :surrender: 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 :banana: 
And the winner is:
k=61574201535, n=80 :banana: :banana: 
For Lars
[QUOTE=ltd]Are n=74,75,76 free ?
I would like to test them. Lars[/QUOTE] You might want to try a bit higher, given the latest record. This is addictive!! Regards Robert Smith 
OK i take 100,101,102
Lars 
Another one for n = 80
k = 632893190475 !! 
n = 80 finished for k < 10^12. On to bigger n's :showoff:

[QUOTE=axn1]Hmmm.... Must be the version. I am using 2.82. If I put k higher than 2^321 it just wont budge :surrender:
[/QUOTE] Strange i use 2.82 also and i have no problems sieving >2^32. 
[QUOTE=ltd]Strange i use 2.82 also and i have no problems sieving >2^32.[/QUOTE]
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. 
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 
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 
no luck either
I did n=110 last night to k=10^11, no octoproths thier either
Regards Robert Smith 
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 
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 :grin: 
1 Attachment(s)
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 :redface: 
[QUOTE=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 :redface:[/QUOTE] May I ask you which boost of performance gave the substitution of Pascal code with asm code? Luigi 
[QUOTE=ET_]May I ask you which boost of performance gave the substitution of Pascal code with asm code?
Luigi[/QUOTE] The two divisions in TestK  gave appr. 35% speedup. Not much but I'll take any speedup especially since I am running these for 23 days at a stretch. Actually, there is one more optimization there  the division of k by p is done by two backtoback 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 nonasm version, you can use suitable qword operations. But in such cases, it might be worthwhile to use alternatives to division. 
[QUOTE=axn1]Results for n = 97 (after 65% completion to k<10^13)
1926973493115 2212009461375 2412877121565 5647136892825 [/QUOTE] One more to the list for n = 97 6832047128535 
1 Attachment(s)
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.

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 
Here's two small ones I found using axn1's program.
8299358445 50 3920165865 54 
And of course as soon as I posted those I found some more...
13419352155 52 14002823745 52 19306888875 52 26648959155 52 
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. 
[QUOTE=axn1]One more to the list for n = 97
6832047128535[/QUOTE] The rest for n = 97: 8246997577755 8883883726185 9417272582445 9910177359165 These are the last for k < 10^13 
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 
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 
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. :rolleyes: TTn 
[QUOTE=robert44444uk]However, how do you write the line script to create an output file? It is years since I saw dos. [/QUOTE]
octo 50 10 > candidates.txt [QUOTE=robert44444uk]I wonder what is so special about 52, it seems statistically well outside of normal variances?[/QUOTE] 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. :whistle: 
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 3PRP! (0.0001s+0.0002s) 374526655755*2^1131 is 3PRP! (0.0001s+0.0045s)  Twin  374526655755*2^(113+1)+1 is 3PRP! (0.0001s+0.0079s) 374526655755*2^(113+1)1 is 3PRP! (0.0001s+0.0045s)  BiTwin  2^113+374526655755 is 3PRP! (0.0030s+0.0002s) 2^113374526655755 is 3PRP! (0.0001s+0.0067s) 2^(113+1)+374526655755 is 3PRP! (0.0001s+0.0042s) 2^(113+1)374526655755 is 3PRP! (0.0001s+0.0042s)  Complete Set  Regards Robert Smith 
Well done robert, they're all prime by the way. However the largest known is
k=405777203685 n=120 found by axn1. 
Small Octoproths
These are the smallest octoproths for their corresponding bases. Why 56 is so large is a real headscratcher.
8299358445 50 106546113135 51 13419352155 52 216800357445 53 3920165865 54 72038479785 55 590925115935 56 138429315465 57 84183246225 58 107884757295 59 
Results for n = 130, (k < 10^13)
1075252753275 3408331609305 7076113724805 
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*2^11 = 29 15*2^(1+1)+1 = 61 15*2^(1+1)1 = 59 2^1+15 = 17 2^115 = 13 2^(1+1)+15 = 19 2^(1+1)15 = 11 If you count negative primes too. 
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^nk 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: [url]http://primes.utm.edu/curios/[/url] Regards Robert Smith 
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 
Smallest
So, if my program works properly, there are no (certified prime) octoproths within the ranges n=3150 and k=1521207165.
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 kvalue 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. 
oct
I working on the new version now...
:cool: 
Some maths
If k*2^n+1 is an octoproth then
k = 1 mod 2. If k is even then 2^(n+1) + k is divisible by 2. k = 0 mod 3. If k = 1 mod 3 then either 3 divides k*2^n+1 or k*2^(n+1)+1. Similarly for k = 2 mod 3 k = 0 mod 5. k = 0 mod 7 or (n = 1 mod 3 and k = +/ 1 mod 7). I can't make any other useful criteria from any other primes. Does anybody know of other goodies like this? 
Completed 31
Only two octoproths for n=31, this base is now complete.
196168335 31 1813059975 31 
[QUOTE=Dougy]So, if my program works properly, there are no (certified prime) octoproths within the ranges n=3150 and k=1521207165.
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 kvalue 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.[/QUOTE] I have verified that there are no octo's between 10 <= n <= 26. Also there are no octo's in the range 3150 for k < 10^7. I am right now in the process of checking whether 21207165 is the smallest possible for n <= 1000 
Weights of certain bases.
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Today I took a look at the number of candidates remaining after running axn1's sieve to 10^10. I ran the sieve over n=50 to n=150.
I will call the "weight" of a base n, to be the number of candidates remaining after running the sieve through 10^10. The number of candidates remaining: Average weight = 18.15 Minimum weight = 3 (n=68) the lightest. Maximum weight = 54 (n=112) the heaviest. Also 8299358445 50 3920165865 54 7130617935 62 925905105 64 3539387145 65 were the only primeoctoproths found. I've attached an excel spreadsheet with the details, and 101 text files with the output from the sieve for n=50 to 100. 
Organised search.
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Firstly it seems that the 'heavy' bases are more likely to produce an octo than the 'light' bases. So n=52, 67, 70, 82, 97, 112, 115, 142, ... (bases with weight >= 40) would be a good place to start searching. With this reasoning (whether sound or not) I discovered:
65498827395 67 :smile: ... :alien: In an attempt to make this search a bit more organised I've created a text file listing what has already been searched. I'll try to update this regularly, and as often as possible. Btw, tell me if I've missed anything or you've searched a region more than what is listed. In this file, a typical base would look like this: <k> <n> <discoverer> <k> <n> <discoverer> ... <k> <n> <discoverer> (searched/2^n) if an octo exists, and <n> (searched/2^n) otherwise. This way we can hopefully not redo others work. PS: If I haven't missed any, we now know of 97 octoprothprimes. Hopefully we'll make the 100 mark soon. :geek: PS2: Most wanted octoprothsprimes: base 32, 63. 
It seems that the program is not sieveing, but trail factoring to a certain prime, am i right?
I've been playing a bit with SMALL_PRIME and MAX_SMALL_PRIME, which does let the program factor further or less deep, (and spit out more or less possible candidates) but surprisingly, it doesn't really finishes a range faster when i factor less deep. What i want is quickly generate an output file which cancels most candidates by factoring to lets say 1000 or 10000 and continueing with newpgen. This makes it easier to stop/resume, to see how many candidates are removed in respect to prping (important when n gets larger) and it might be faster. 
[QUOTE=smh]It seems that the program is not sieveing, but trail factoring to a certain prime, am i right?[/QUOTE]
That's correct. However, the at the price of about 3 modulo calculations per prime you can test all 8 forms. [QUOTE=smh]I've been playing a bit with SMALL_PRIME and MAX_SMALL_PRIME, which does let the program factor further or less deep, (and spit out more or less possible candidates) but surprisingly, it doesn't really finishes a range faster when i factor less deep.[/QUOTE] Since the majority of candidates get sieved out (or trial factored out) within a handful of primes, only a few will be tested by the higher primes. Again, it all comes back to all 8 forms being simultanously checked. [QUOTE=smh]What i want is quickly generate an output file which cancels most candidates by factoring to lets say 1000 or 10000 and continueing with newpgen. This makes it easier to stop/resume, to see how many candidates are removed in respect to prping (important when n gets larger) and it might be faster.[/QUOTE] There is no real reason to run the sieve output thru NewPGen. The only improvement that I can think of is to augment the trial division approach with a sieve for the first few primes. I have no idea how much improvement that'll make. However, it still will be better to trial divide to higher primes than use NewPGen (4 forms Vs 8 forms = big difference!). Also, if you are searching higher n's, the chance of finding an octo is *really* low! 
Scarce
So it looks like octos are scarce, but I am wondering if, like twins, they are infinite?
There are a finite number of k to check for each n, and given that each n only multiples the candidates to check by 2, and given the prime occurence rule, 1/ln(x), which has to be scaled by a factor of between 4 and 8 (k.2^n is bigger than 2^nk), then, we should be able to sigma a formula. I am not a mathematician, and I know there are big dangers around playing with inifinity and converging and diverging series, let alone proving anything. If either case you might imagine that there may be dodecaproths (whatever), which are 3 chains, I imagine these might be enormously rare and maybe nonexistent if there are only finite octos. Interesting challenge that to prove that as well! If anyone proves it and writes a paper I want a credit so that I might get an Erdos number!!!! Regards Robert Smith 
[QUOTE=axn1]Since the majority of candidates get sieved out (or trial factored out) within a handful of primes, only a few will be tested by the higher primes. Again, it all comes back to all 8 forms being simultanously checked.
[/quote] Thanks, that makes sence, didn't think of that. [QUOTE=axn1]There is no real reason to run the sieve output thru NewPGen. .........it still will be better to trial divide to higher primes than use NewPGen (4 forms Vs 8 forms = big difference!). [/quote] Probably true, the reason i asked is because i quickly wanted to get a file with (a lot) of candidates, instead of waiting a long time before all candidates are tested. Time on a prp test is short with these low n's anyway. [QUOTE=axn1]Also, if you are searching higher n's, the chance of finding an octo is *really* low![/QUOTE] Isn't that what makes them more special? 
New and improved!
1 Attachment(s)
New and improved sieve !
3x to 8x speed up (depending on how deep you sieve). You can sieve specific ranges  start and end values for k. Three executables  fast, medium and deep sieves (p = 10^5, 10^6 and 10^7 resp.) 
Troubles with the new program.
I've had both success and troubles with the new software. Firstly
22573117995 37 23055820515 37 45547390455 37 66695049135 37 73828108635 37 78745654785 37 108269914095 37 132750210165 37 136640238735 37 were discovered using the new program. :banana: However I tested the program on the known n=35 with 231235935 and 422362395 being the only octoproths. Unfortunately it sieved out the latter. (all three versions did it). :sad: Furthermore often it seems that terms with small factors (7, 11, 13) are not sieved. :unsure: For example the following remained. 32803605*2^56+1 = 4884169 * 483961315030365049 32803605*2^561 = 7 * 13 * 25975262110665308033069 32803605*2^(56+1)+1 = 4727497704141086062018561 32803605*2^(56+1)1 = 8220335593 * 575097896023463 2^56+32803605 = 72057594070731541 2^5632803605 = 72057594005124331 2^(56+1)+32803605 = 17147939 * 8404227943 2^(56+1)32803605 = 22878377 * 6299187571 
[QUOTE=Dougy]I've had both success and troubles with the new software. Firstly
22573117995 37 23055820515 37 45547390455 37 66695049135 37 73828108635 37 78745654785 37 108269914095 37 132750210165 37 136640238735 37 were discovered using the new program. :banana: However I tested the program on the known n=35 with 231235935 and 422362395 being the only octoproths. Unfortunately it sieved out the latter. (all three versions did it). :sad: Furthermore often it seems that terms with small factors (7, 11, 13) are not sieved. :unsure: For example the following remained. 32803605*2^56+1 = 4884169 * 483961315030365049 32803605*2^561 = 7 * 13 * 25975262110665308033069 32803605*2^(56+1)+1 = 4727497704141086062018561 32803605*2^(56+1)1 = 8220335593 * 575097896023463 2^56+32803605 = 72057594070731541 2^5632803605 = 72057594005124331 2^(56+1)+32803605 = 17147939 * 8404227943 2^(56+1)32803605 = 22878377 * 6299187571[/QUOTE] Hmmm.. That shouldn't happen :no: I'll look into it. 
More newies...
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610118148045 41
802757470515 41 832494696285 41 
Bug Fix
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There was an unnecessary increment of k inside the loop. Fixed it. Can someone test it and confirm ?
@Dougy  You might want to redo the searches done with the last version :sad: 
Using the new version of axn1's program, I found the following octoproths for k=235 until n=3000000000000 (=3*10^12):
562223326335 235 722744559915 235 926010118305 235 2441346583515 235 I'm looking further on k=235! One question: what's the difference between the 3 sieve executables? I'm currently using octo_deep.exe. 
[QUOTE=Templus]One question: what's the difference between the 3 sieve executables? I'm currently using octo_deep.exe.[/QUOTE]
They differ only in the depth of the sieve, ie, the number of p's used to sieve. p < 10^5, 10^6 and 10^7 (resp. for fast, med & deep). PS: The numbers you posted are not octos. They are only prime for 2^n+/k and 2^(n+1)+/k. They are not prime for the other forms, k*2^n+/1 and k*2^(n+1)+/1 
I tested n=141 with the old executable upto k=5,555*10^13
This is the closest i got: [CODE]43354856050725*2^141+1 is 3PRP! (0.0004s+0.0004s) 43354856050725*2^1411 is 3PRP! (0.0002s+0.0038s) 43354856050725*2^(141+1)+1 is 3PRP! (0.0002s+0.0022s) 43354856050725*2^(141+1)1 is 3PRP! (0.0002s+0.0021s) 2^141+43354856050725 is 3PRP! (0.0001s+0.0021s) 2^14143354856050725 is 3PRP! (0.0001s+0.0030s) 2^(141+1)+43354856050725 is 3PRP! (0.0001s+0.0023s) 2^(141+1)43354856050725 is composite: [2FAAB440A92EB3DC] (0.0001s+0.0022s)[/CODE] I'll try 135 next 
[QUOTE=smh]I'll try 135 next[/QUOTE]
If you are searching higher n's, try to search n = 4,7, or 10 (mod 15), esp n = 7 (mod 15). These are heavy weight n's. I think n=142 will make a good candidate. 
[QUOTE=axn1]They differ only in the depth of the sieve, ie, the number of p's used to sieve. p < 10^5, 10^6 and 10^7 (resp. for fast, med & deep).
PS: The numbers you posted are not octos. They are only prime for 2^n+/k and 2^(n+1)+/k. They are not prime for the other forms, k*2^n+/1 and k*2^(n+1)+/1[/QUOTE] I am terribly sorry, I forgot to extend the ABCline :redface: :redface: I will test k=235 further on! 
[QUOTE=axn1]If you are searching higher n's, try to search n = 4,7, or 10 (mod 15), esp n = 7 (mod 15). These are heavy weight n's. I think n=142 will make a good candidate.[/QUOTE]
Okay, i'll test 142 then. I'm running the 'sieve' for a day or so on a p3 700 and see what pops up. For 135 i didn't find any upto 1.196*10^13 12 had primes for the first 6 forms, of which 2 had also primes for the 7th form 
Wow 37!!
So far every test I have put the new program through was passed. So I believe it is working fine.
I ran the complete test on bases upto n=37. (and still going). And that base produced a whopping 83 octoproth primes. :showoff: Secondly, both bases 32 and 33 have no octoproth primes. :sad: 
Hmmm a bug in Dario Alpern's ECM
When I put these (and others) into the batch factorisation:
2^39540206575755 2^39539552526135 and test for primality it says (even if i just type in the decimal too!) 9549238133 is composite 10203287753 is composite However if I factorize them instead 9549238133 = 9549238133 10203287753 = 10203287753 implying they're prime. This means that I will have missed some octoproths... :sad: But fortunately I kept the sieved files. URL: [url]http://www.alpertron.com.ar/ECM.HTM[/url] 
[QUOTE=Dougy]When I put these (and others) into the batch factorisation:
2^39540206575755 2^39539552526135 and test for primality it says (even if i just type in the decimal too!) 9549238133 is composite 10203287753 is composite However if I factorize them instead 9549238133 = 9549238133 10203287753 = 10203287753 implying they're prime. This means that I will have missed some octoproths... :sad: But fortunately I kept the sieved files. URL: [url]http://www.alpertron.com.ar/ECM.HTM[/url][/QUOTE] I too ran into this problem yesterday, while working with one of the lower n's (32, I think). The "deep" version sieves upto p < 10^7, which means that for n <= 45 all the candidates will be automatically prime for 2^n+/k forms! You definitely need to recheck base 32 and 33! 
Interesting things...
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After rechecking the small bases, I've updated the text file again. It should be fixed. I've completed upto n=41. Some interesting properties...
Number of octoprothprimes for n=27,28,29,... 1,2,1,1,2,0,0,7,17,11,90,28,83,331,109,... n=40 alone has 331 octoprothprimes :exclaim: To think only the other day I was hoping to break the 100 mark. Number of kvalues unsieved (octo_deep) for n=27,28,29,... over all possible kvalues. 2,4,4,2,19,22,13,110,137,85,802,360,844,4434,1651,7552,... 
[QUOTE=Dougy]After rechecking the small bases, I've updated the text file again. It should be fixed. I've completed upto n=41. Some interesting properties...
Number of octoprothprimes for n=27,28,29,... 1,2,1,1,2,0,0,7,17,11,90,28,83,331,109,... n=40 alone has 331 octoprothprimes :exclaim: To think only the other day I was hoping to break the 100 mark. Number of kvalues unsieved (octo_deep) for n=27,28,29,... over all possible kvalues. 2,4,4,2,19,22,13,110,137,85,802,360,844,4434,1651,7552,...[/QUOTE] Some missing values for n=32,33,34,35, and 37. [code]n = 32  409668105 664495755 2368386195 2709707805 3383804865 3692088225 3762658725 n = 33  715414875 6876947175 n = 34  293705775 1183281975 1397861655 3767954715 4597935705 8596001505 n=35  17182250085 17783238795 20646922695 21811399155 22622064465 23416146075 24115395465 24449183535 25380028905 n=37  7218568995 126139443165[/code] n = 36,38,39, and 40 are fine. Also, I have checked all high k's for n =27 thru 64 (high k's are k's that can't be reliably sieved because 2^nk becomes smaller than the largest sieve prime). There are no hidden octo's in there :smile: 
Wow lots of holes.
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Wow thanks, I didn't think there could be so many missing... it's all updated now. :smile:
So there might be octo's for all bases after 27. I've added the smallest octo for each base upto 71. :bounce: Most Wanted: n=72... searched k<1000000000000. :question: 
142
I've found the following for n=142, searching K from 1 to 3.09x10^13
[CODE]8444737373415*2^142+1 is 3PRP! (0.0002s+0.0002s) 8444737373415*2^1421 is 3PRP! (0.0002s+0.0023s) 8444737373415*2^(142+1)+1 is 3PRP! (0.0002s+0.0023s) 8444737373415*2^(142+1)1 is 3PRP! (0.0018s+0.0023s) 2^142+8444737373415 is 3PRP! (0.0001s+0.0024s) 2^1428444737373415 is 3PRP! (0.0001s+0.0023s) 2^(142+1)+8444737373415 is 3PRP! (0.0001s+0.0023s) 2^(142+1)8444737373415 is 3PRP! (0.0001s+0.0024s) 9532236817845*2^142+1 is 3PRP! (0.0002s+0.0029s) 9532236817845*2^1421 is 3PRP! (0.0002s+0.0024s) 9532236817845*2^(142+1)+1 is 3PRP! (0.0002s+0.0024s) 9532236817845*2^(142+1)1 is 3PRP! (0.0002s+0.0025s) 2^142+9532236817845 is 3PRP! (0.0001s+0.0023s) 2^1429532236817845 is 3PRP! (0.0001s+0.0023s) 2^(142+1)+9532236817845 is 3PRP! (0.0001s+0.0023s) 2^(142+1)9532236817845 is 3PRP! (0.0001s+0.0026s) 22732824274545*2^142+1 is 3PRP! (0.0002s+0.0003s) 22732824274545*2^1421 is 3PRP! (0.0002s+0.0024s) 22732824274545*2^(142+1)+1 is 3PRP! (0.0002s+0.0024s) 22732824274545*2^(142+1)1 is 3PRP! (0.0002s+0.0024s) 2^142+22732824274545 is 3PRP! (0.0001s+0.0023s) 2^14222732824274545 is 3PRP! (0.0001s+0.0023s) 2^(142+1)+22732824274545 is 3PRP! (0.0001s+0.0024s) 2^(142+1)22732824274545 is 3PRP! (0.0001s+0.0024s)[/CODE] :banana: :bounce: :banana: Close to 2 million numbers survived the sieve. Newpgen didn't make sence after this, since it removed candidates much slower than i was able to prp them. I'll try 157 (another 7 mod 15) next. 
Yay!
:w00t: Fantastic find! They're all primes too. Three new largest octoproths.

Found the most wanted
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Yay!
1689898522725 72 1704984216915 72 1729280652225 72 1993468873725 72 The new most wanted is now n=74 (425575968195/18889465931478580854784 searched) Oh and I (finally) completed n=42. :sleep: 
Compression
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Because there were too many octo's with small bases I've zipped up the complete list of octo's for bases n<=42.

Octo RMA 1.74 delay
I've been sick lately, and have not made much progress with octo code, but I know now, exactly how it will be integrated.
Great work 15k'ers! TTn 
[QUOTE=Dougy]:w00t: Fantastic find! They're all primes too. Three new largest octoproths.[/QUOTE]
I knew i forgot something. I didn't test them for primality. Thanks for doing so. :bow: 
New most wanted n=87
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I've been trying to find the smallest octos for each base. Lots of success... however n=87 is a stubborn base. Sieved upto 4*10^12, and no success.
Bedtime now. :yawn: 
[QUOTE=Dougy]I've been trying to find the smallest octos for each base. Lots of success... however n=87 is a stubborn base. Sieved upto 4*10^12, and no success.
Bedtime now. :yawn:[/QUOTE] Okay, i have a few hours sieving time available, i'll search a bit further on this K. FYI, i searched n=142 to 3.09x10^13 
[QUOTE=smh]Okay, i have a few hours sieving time available, i'll search a bit further on this K.
FYI, i searched n=142 to 3.09x10^13[/QUOTE] May I ask: With so many other established computational projects already available, why do this one? I accuse noone. However, I suspect that the answer is: "Because this project is new, it is relatively easy to get results". Let me quote John Kennedy: "We choose to go to the moon and do the other thing... Not because they are easy, but because they are hard." There is little reward in using other people's software to do easy computations. Rewards come from succeeding at something that is hard. The satisfaction one derives from solving a problem is commensurate with the level of effort. :unsure: 
[QUOTE=R.D. Silverman]
There is little reward in using other people's software to do easy computations. Rewards come from succeeding at something that is hard. The satisfaction one derives from solving a problem is commensurate with the level of effort. :unsure:[/QUOTE] Agreed, but an important question is "hard for whom?" There is little reward in failing to make progress because one doesn't have the resources available to complete the task. Sometimes relatively easy tasks are best attempted by people with relatively lowly resources. They free up the people with massive resources from having to clear them up for completeness sake. That is why I am factoring 6,768L.c130 personally rather than using the grossly disproportionate resources of NFSNET. I leave open completely the question of whether everyone has the same value function. Paul 
n = 44 complete
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I am doing n=45 now.

[QUOTE=xilman]Agreed, but an important question is "hard for whom?" There is little reward in failing to make progress because one doesn't have the resources available to complete the task.
Sometimes relatively easy tasks are best attempted by people with relatively lowly resources. They free up the people with massive resources from having to clear them up for completeness sake. That is why I am factoring 6,768L.c130 personally rather than using the grossly disproportionate resources of NFSNET. I leave open completely the question of whether everyone has the same value function. Paul[/QUOTE] And please don't forget that those results come from axn1 software. I'm sure he studied hard to get the most from it, and this project is a reward to his effort. The results may (maybe) be easily achieved from other searches or software, but there is little advancement in personal knowledge from using others' programs; instead he, I guess, learnt a lot to code it and gave other people the hint to search and deepen the subject by themselves; perhaps he also gave others the will to try and learn by themselves subjects otherwise not chosen. Luigi 
[QUOTE=R.D. Silverman]May I ask:
With so many other established computational projects already available, why do this one?[/QUOTE] This one is as good as any other. But i choose this one because i can't use that pc for anything else (except maybe ecm) and it's only available to me for a few days. PRPing takes only 15 minutes on my P4. The pc will be gone tomorrow morning and i didn't want to let it run idle. For the last year or two, i spent most of my cpu time factoring for a few projects (including Paul's). Unfortunately, Cunningham numbers are a bit to hard for my limited resources. I might be able to do a < c140 with GNFS, but my P4 has only 256MB RAM. I have a laptop with 2GB, but that one is used for work, and part of that memory is often needed. Besides, i'm not sure GGNFS is upto the job yet. Sieving on multiple pc's requires to much book keeping for the time i've currentely got. I'm open for any number you want to have factored, as long as it can be done in a reasonable amount of time. 
N=157
Below the results for N=157 and K<2*10^13 :banana:
All are prp, not tested for primality [code]541364899635*2^157+1 541364899635*2^1571 541364899635*2^(157+1)+1 541364899635*2^(157+1)1 2^157+541364899635 2^157541364899635 2^(157+1)+541364899635 2^(157+1)541364899635 5375998941495*2^157+1 5375998941495*2^1571 5375998941495*2^(157+1)+1 5375998941495*2^(157+1)1 2^157+5375998941495 2^1575375998941495 2^(157+1)+5375998941495 2^(157+1)5375998941495 7137839620995*2^157+1 7137839620995*2^1571 7137839620995*2^(157+1)+1 7137839620995*2^(157+1)1 2^157+7137839620995 2^1577137839620995 2^(157+1)+7137839620995 2^(157+1)7137839620995 16986089468655*2^157+1 16986089468655*2^1571 16986089468655*2^(157+1)+1 16986089468655*2^(157+1)1 2^157+16986089468655 2^15716986089468655 2^(157+1)+16986089468655 2^(157+1)16986089468655 [/code] 
Silverman
:nuke:
[QUOTE]May I ask: With so many other established computational projects already available, why do this one? I accuse noone. However, I suspect that the answer is: "Because this project is new, it is relatively easy to get results". Let me quote John Kennedy: "We choose to go to the moon and do the other thing... Not because they are easy, but because they are hard." There is little reward in using other people's software to do easy computations. Rewards come from succeeding at something that is hard. The satisfaction one derives from solving a problem is commensurate with the level of effort. [/QUOTE] I would appreciate if you respectfully ask questions elsewhere. (buzz off) This project has been around for awhile, and now has a new spin on the fundamental property that started it. 
Angels
We would still be computing the number of angels on the head of a pin unless we had adapted and explored.
The facts are: This is original thinking, adaptation and exploration in virgin territory It is simple, but not so simple, there are pitfalls for the unwary There is interesting mathematics if anyone wants to look at it Axn1's program is a masterpiece This is true 1+1=3, as I like to define human cooperation This is the shitzit right now Mr Silverman, have pity on us poor humans, nattering together and taking up your valuable bandwidth. Consider yourself flamed, or at least singed, as I am a pacific sort. Regards Robert Smith (aged 53 1/4) 
What's in a name?
Silverman? Hmmm... sounds familiar. Are you related to Joseph H. Silverman who published (among others) the paper "Wieferich's Criterion and the abcConjecture" in the Journal of Number Theory?

Predicting the next octoproth.
I've been looking at the "weights" of certain bases again. Working as I defined it previously, W(157) = 62. Is the heaviest W(n) for n <= 200.
The next three heaviest are :showoff: W(175) = 60 W(187) = 56 W(112) = 54 So the bases n = 112, 175, 187 should be good for finding octos. 
Update
Newies:
4358737887315 87 (a monster!) :exclaim: 2232065722095 88 2148136610235 88 for n=8996 there are no known octoproths. Searched n=89 to k=780321515295. 
bitwin
[QUOTE]Okay, i have a few hours sieving time available[/QUOTE]
Is'nt it faster to verify bitwin primality first? With RMA 1.75 in 2 minutes I was able to find no octoproth's for n=5000, with k<200000000. TTn 
[QUOTE=TTn]Is'nt it faster to verify bitwin primality first?[/QUOTE]
No, the remaining 4 possibilities often have small factors. [QUOTE=TTn]With RMA 1.75 in 2 minutes I was able to find no octoproth's for n=5000, with k<200000000. [/QUOTE] With octo_fast it took less then 1 second to 'sieve' this range, with only 4 possible candidates left. [CODE]1297905 5000 111183135 5000 116381265 5000 176976555 5000[/CODE] 
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