Dodecaproths

[robert44444uk]

Check me if I am wrong, but I ran the first 65000 octoproths through my spider routine and turned up the following dodecaproths (3-chain):

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 n-1, 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

[robert44444uk]

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

[Kosmaj]

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:

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!!

where I define (k,n) to be DodecaProth if both (k,n) and (k,n+1) are OctoProths. BTW, I checked the table provided by Greenbank but found no DodecaProths for n=50.

[Kosmaj]

Found 3 DodecaProths for n=51

145174433549145 51 is OctoProth! ... and DodecaProth!!
246834311745945 51 is OctoProth! ... and DodecaProth!!
868049887559295 51 is OctoProth! ... and DodecaProth!!

No more legs neither to the left (n=50) nor to the right (n=54).

[R. Gerbicz]

Originally Posted by Kosmaj

I define (k,n) to be DodecaProth if both (k,n) and (k,n+1) are OctoProths.

What a nice definition!
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!

[Kosmaj]

Sometime before I thought to define ....proth ( what is the correct name for sixteen? )

I think it should be called HexadecaProth, and the next one if we ever encounter it is IcosaProth

It means 16 primes, but today to find such a pattern is impossible, even by a fast sieve and fast prp checking.

I don't think it's impossible! We have to find enough (i.e., many!) DodecaProths and one of them will surely be HexadecaProth. Check the following chain of 16 primes (the so-called bitwin with 7 links):
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!

[R. Gerbicz]

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 ).

[R. Gerbicz]

Originally Posted by Kosmaj

This is I think the place to look for HexadecaProths, around our exponent n=65 or 70.

Sieving for Hexadecaproth would be faster than my new Dodeca program.
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.

[R. Gerbicz]

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

[Kosmaj]

You are so fast! Thanks for the new program.

So by this program you won't able to find a Hexadecaproth.

No problems, let's first see how many DodecaProths can we find. For some n's most likely we'll have to go all the way to k=1E17 or more. BTW, using your new program I confirmed those for n=49 and n=51 and now I'm trying n=56 (because we can trivially locate those for n=52-55 when the guys find all OctoProths). Up to k=1000T (1E15) nothing...

[R. Gerbicz]

Originally Posted by Kosmaj

You are so fast! Thanks for the new program.

I haven't written that program between 14 Jan 06 03:39 PM and 03:51 PM. ( See the times. ) I have started the writing some hours before, but still today, it wasn't very hard.

Originally Posted by Kosmaj

No problems, let's first see how many DodecaProths can we find. For some n's most likely we'll have to go all the way to k=1E17 or more. BTW, using your new program I confirmed those for n=49 and n=51 and now I'm trying n=56 (because we can trivially locate those for n=52-55 when the guys find all OctoProths). Up to k=1000T (1E15) nothing...

When you are checking this I hope you are using kmin=1 and kmax=-1+2^n