Thread: Hexadecaproths
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Old 2006-01-19, 16:18   #4
R. Gerbicz
 
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"Robert Gerbicz"
Oct 2005
Hungary

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Quote:
Originally Posted by Greenbank
No program exists (yet) to search for them, but I'm sure it will soon...
Yes. There will be a program, probably 2 or 3 weeks, I've other tasks now. But I think that it would be a little different than octo or dodeca program, because it's computation time is ( much ) larger. Here it is my suggestion:
We could start a distributed search for n=76, because for this number the expected number of hexadecaproths is about 16.
Weight of n ( for hexadecaproth ) is ( PARI ):
Code:
w(n)=T=32768.0;forprime(p=3,10^4,l=listcreate(16);g=Mod(2,p)^n;h=1/g;\
a=[g,-g,h,-h,2*g,-2*g,h/2,-h/2,4*g,-4*g,h/4,-h/4,8*g,-8*g,h/8,-h/8];\
a=lift(a);for(i=1,16,listput(l,a[i],i));l=listsort(l,1);\
T*=(1-length(l)/p)/(1-1/p)^16);return(T)
Then we can esimate the number of hexadecaproths for a given n by:
Code:
f(n)=round(w(n)*2^n/(n*log(2))^16*1/256)
The first value that f(n)>0 is n=71:
f(71)=1
f(72)=0
f(73)=0
f(74)=1
f(75)=2
f(76)=16
f(77)=1
f(78)=1
f(79)=15
Probably we could search for n=71, but it seems much better n=76, and we can also find some dodecaproth as we search for hexadecaproth, but not all of them, because the program will optimized for hexadecaproth search.

The expected running time to find one! hexadecaproth for n=76 is about 2 years for my PC ( 1.7 GHz Celeron ) ( if my calculation is correct, note that this is only true to find one hexadecaproth! for n=76, not for all of them, to find all of them the expected time is 16 times larger). So we can find hexadecaproth!
I imagine this that there will be "workunits", so you can complete one workunit in about half an hour on an average computer. In one workunit the computer examine all possible cases in one remainder class modulo T. So the search is not an interval search like in octo-dodeca program.
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