20050415, 04:26  #45 
Jun 2003
5,081 Posts 
Results for n = 130, (k < 10^13)
1075252753275 3408331609305 7076113724805 
20050415, 08:38  #46 
Aug 2004
Melbourne, Australia
2^{3}·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*2^11 = 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. 
20050415, 15:38  #47 
Jun 2003
Oxford, UK
5·389 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^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: http://primes.utm.edu/curios/ Regards Robert Smith 
20050415, 15:45  #48 
Jun 2003
Oxford, UK
5×389 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 
20050416, 01:20  #49 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
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. 
20050416, 03:30  #50 
2·23·197 Posts 
oct
I working on the new version now...

20050416, 06:50  #51 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
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? 
20050416, 08:43  #52 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
Completed 31
Only two octoproths for n=31, this base is now complete.
196168335 31 1813059975 31 
20050416, 13:56  #53  
Jun 2003
5,081 Posts 
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 

20050417, 08:43  #54 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
Weights of certain bases.
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. Last fiddled with by Dougy on 20050417 at 08:44 
20050417, 12:25  #55 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
Organised search.
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 ... 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. PS2: Most wanted octoprothsprimes: base 32, 63. 
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