20060113, 19:40  #1 
Jun 2003
Oxford, UK
2^{2}×13×37 Posts 
Dodecaproths exist!
Check me if I am wrong, but I ran the first 65000 octoproths through my spider routine and turned up the following dodecaproths (3chain):
44 9604223498415 1 4 5 47 45960089776965 1 4 5 49 374180855930805 1 4 5 44 10872870991605 0 4 4 44 7946515823715 0 4 4 47 69283546229205 0 4 4 49 19000157002995 0 4 4 49 502414540060965 0 4 4 49 555428994253665 0 4 4 45 9604223498415 4 1 5 45 7946515823715 4 0 4 45 10872870991605 4 0 4 48 69283546229205 4 0 4 48 45960089776965 4 0 4 The firs column is n, the second is k, the third is the four types, with n1, the fourth column is the four types with n+2, and the final column the number of legs on the spider. So we have some 5's but no 6's. They maybe hope for a 4 chain yet!!! Regards Robert Smith 
20060113, 19:50  #2 
Jun 2003
Oxford, UK
2^{2}·13·37 Posts 
Statistics
Some stats relating to ther frequency of "legs" in my test:
lower legs upper legs total legs 0 53444 81.549% 49382 75.351% 40477 61.763% 1 11031 16.832% 14524 22.162% 19846 30.283% 2 1012 1.544% 1513 2.309% 4459 6.804% 3 44 0.067% 108 0.165% 673 1.027% 4 5 0.008% 9 0.014% 71 0.108% 5 0 0.000% 0 0.000% 10 0.015% 65536 65536 65536 Regards Robert Smith 
20060114, 08:44  #3 
Nov 2003
2·1,811 Posts 
Re: Dodecaproths
Robert, congrats on locating first dodecaproths. But note that there are 9 not 14 distinct ones because you counted some twice (once to the right, once to the left) like for example:
44 9604223498415 1 4 5 45 9604223498415 4 1 5 I verified that members of all 9 are primes, here is the output of my Pari script: Code:
7946515823715 44 is OctoProth! ... and DodecaProth!! 9604223498415 44 is OctoProth! ... and DodecaProth!! 10872870991605 44 is OctoProth! ... and DodecaProth!! 45960089776965 47 is OctoProth! ... and DodecaProth!! 69283546229205 47 is OctoProth! ... and DodecaProth!! 19000157002995 49 is OctoProth! ... and DodecaProth!! 374180855930805 49 is OctoProth! ... and DodecaProth!! 502414540060965 49 is OctoProth! ... and DodecaProth!! 555428994253665 49 is OctoProth! ... and DodecaProth!! 
20060114, 09:33  #4 
Nov 2003
2×1,811 Posts 
n=51
Found 3 DodecaProths for n=51
Code:
145174433549145 51 is OctoProth! ... and DodecaProth!! 246834311745945 51 is OctoProth! ... and DodecaProth!! 868049887559295 51 is OctoProth! ... and DodecaProth!! 
20060114, 10:23  #5  
"Robert Gerbicz"
Oct 2005
Hungary
2·7·103 Posts 
Quote:
Probably it is also possible to write a GMP program for this search, like for octoproth. Sometime before I thought to define ....proth ( what is the correct name for sixteen? ), where (k,n) and (k,n+2) are Octoproths ( this means that this is also a DodecaProth because it is easy to prove that if (k,n) and (k,n+2) are Octoproths then (k,n+1) is also an octoproth ). It means 16 primes, but today to find such a pattern is impossible, even by a fast sieve and fast prp checking. Ps. Searching for Dodecaproth would be faster for a given n value and a given Range than searching for Octoproths. You don't need to find all octoproths in that range! Last fiddled with by R. Gerbicz on 20060114 at 10:31 

20060114, 14:39  #6  
Nov 2003
111000100110_{2} Posts 
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517144126484002331*13#*2^n+/1 n=0,1,2,3,4,5,6,7; and "#" denotes primorial. composed of 8 twins, and two Cunningham chains of first and second kind. Have you ever thought that such a chain exists? Only two such chains are currently known. The twin for n=0 has 23 digits. This is I think the place to look for HexadecaProths, around our exponent n=65 or 70. If you can change your program to search for DodecaProths (ignoring of course potenital OctoProths whose "right legs" have small factors) I'll be willing to try! Last fiddled with by Kosmaj on 20060114 at 14:43 

20060114, 14:51  #7 
"Robert Gerbicz"
Oct 2005
Hungary
2×7×103 Posts 
dodeca program version 1.0
I've finished a new program. It is searching for dodecaproths, and it is faster than octoproths!!!
It is sieving all 12 numbers. The speed up is very very different for different ranges and n values. Here I sieve only up to 32000 ( so I can use unsigned short variables ) and magic_constant=32000 The size of used memory is about 130 KB. The number of Prp tests is also much much smaller. You can find the results also in results_dodeca.txt file. Exe for windows: http://www.robertgerbicz.tar.hu/dodeca_1_0.exe Or see the attachment for the code. It isn't very easy to test because we know only a very few of dodecaproths, but for n=44 and n=47 it is correct. It would be good to test this program. Greenbank: I would like to see a new thread: Number of dodecaproths per n Using your tables it isn't hard to create this table. It is easier to find this number than octoproth's number, so maybe we can go further n=55, perhaps n=56, but before this check my program. I've disabled to use n>99, because in this case the expected smallest dodecaproth's k value is very large ( larger than 10^20>2^60=kmax limit ). Last fiddled with by R. Gerbicz on 20060114 at 15:06 
20060114, 16:27  #8  
"Robert Gerbicz"
Oct 2005
Hungary
2·7·103 Posts 
Quote:
Because it is also possible to sieve 16 forms at once!!! I've calculated that the smallest n value for that there exist a Hexadecaproth is probably n=71 But in this case the smallest k value is also very large, say about 2^70, and it is much larger than 2^60=k max limit. So by this program you won't able to find a Hexadecaproth. 

20060114, 16:45  #9 
"Robert Gerbicz"
Oct 2005
Hungary
2·7·103 Posts 
new dodecaproth for n=52 !!!
I've found this by dodeca_1_0.exe The 12 numbers are primes.
2808528662035845 52 I'm still running n=52, if Greenbank finish all octoproths for n=52 then it will be a quick and good check to see if my new program is good or not. Ps. I've calculated that the expected number of dodecaproths for n=52 is 2 Last fiddled with by R. Gerbicz on 20060114 at 16:49 
20060114, 16:52  #10  
Nov 2003
7046_{8} Posts 
You are so fast! Thanks for the new program.
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20060114, 17:00  #11  
"Robert Gerbicz"
Oct 2005
Hungary
10110100010_{2} Posts 
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