My Math Forum Primitive root of known Wilson Primes

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 October 23rd, 2009, 12:30 PM #1 Member   Joined: Jan 2009 Posts: 95 Thanks: 0 Primitive root of known Wilson Primes The known Wilson Primes are: 5, 13, 563 http://en.wikipedia.org/wiki/Wilson_prime And their Primitive Roots are: PrimitiveRoot(5)=2 PrimitiveRoot(13)=2 PrimitiveRoot(563)=2 http://www.brynmawr.edu/math/people/...primitive.html My knowledge in Number Theory has many holes and P.R. is one of them (or maybe it is a unique black hole where you can travel through the galaxies), and I wonder: 1) ¿Can the remaining Wilson Primes have a P.R of 2? 2) Are there formulas to calculate P.R. for some set of integers. (this is for studying this kind of proofs)
 October 23rd, 2009, 08:57 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Primitive root of known Wilson Primes The primitive roots mod 5 are 2 and 3. The primitive roots mod 13 are 2, 6, 7, and 11. The primitive roots mod 563 are 2,5,6,8,14,15,18,20,22,24,26,29,31,32,34,35,37,38, 41,42,43,45,46,50,53,54,55,56, 60,65,66,72,73,78,79,80,83,85,87,88,89,93,94,95,96 ,97,98,102,104,105,109,111,114 ,115,116,118,122,123,124,125,126,128,129,131,134,1 35,136,138,139,140,142,148,150 ,151,152,154,157,159,162,163,164,165,167,168,172,1 73,180,182,184,195,198,199,200 ,202,203,206,212,214,216,217,219,220,224,226,227,2 29,233,234,235,237,238,239,240 ,242,245,249,254,255,259,260,261,263,264,266,267,2 74,279,282,283,285,286,287,288 ,290,291,292,293,294,295,298,301,305,306,307,310,3 11,312,313,315,316,317,319,320 ,322,327,331,332,333,335,338,340,341,342,345,348,3 50,352,353,354,355,356,358,359 ,362,366,367,369,370,371,372,373,374,375,376,377,3 78,380,382,384,385,386,387,388 ,389,392,393,394,397,402,403,405,407,408,410,414,4 16,417,418,419,420,422,426,430 ,431,433,436,442,443,444,446,450,451,453,455,456,4 57,460,462,463,464,471,472,473 ,477,479,481,482,486,487,488,489,492,493,494,495,4 96,499,500,501,502,504,505,506 ,511,512,514,515,516,519,523,524,527,530,533,535,5 36,538,540,542,544,546,547,550 ,551,552,553,554,556,559,560 Pari/GP: Code: pr(p)=my(f=factor(p-1)[,1]);for(n=2,p-1,if(W(n,f,p),print1(n","))) W(n,f,p)=for(i=1,#f,if(Mod(n,p)^(p\f[i])==1,return(0)));1
 October 23rd, 2009, 10:48 PM #3 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: Primitive root of known Wilson Primes CRGreathouse, you are not human. You do not respond in a human-like manner. I think you are a machine.
October 23rd, 2009, 11:18 PM   #4
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Re: Primitive root of known Wilson Primes

Quote:
 CRGreathouse, you are not human. You do not respond in a human-like manner. I think you are a machine
The proper conjecture for HAL 9000 (http://es.wikipedia.org/wiki/HAL_9000) is that the least Primitive Root for Wilson Primes is 2

And also:

If p is an odd prime P.R. of a Wilson Prime then p can not be P.R. of another distinct Wilson Prime.

I´ve found an article about this topic, but I must study it in deep because I do not understand the Primitive Root formulas and many other things:

A SEARCH FOR WIEFERICH AND WILSON PRIMES
RICHARD CRANDALL, KARL DILCHER, AND CARL POMERANCE

http://math.dartmouth.edu/~carlp/PDF/paper111.pdf

October 24th, 2009, 06:53 AM   #5
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Re: Primitive root of known Wilson Primes

Quote:
 Originally Posted by cknapp CRGreathouse, you are not human. You do not respond in a human-like manner. I think you are a machine.
That does not compute. Abort, retry, fail?

(Love the .sig, by the way.)

October 24th, 2009, 09:15 AM   #6
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Re: Primitive root of known Wilson Primes

Quote:
 Originally Posted by CRGreathouse That does not compute. Abort, retry, fail?
retry.

Quote:
 (Love the .sig, by the way.)
Yeah. This semester has been a math professor quote gold mine...

October 26th, 2009, 10:59 AM   #7
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Re: Primitive root of known Wilson Primes

Quote:
 Originally Posted by Barbarel The proper conjecture for HAL 9000 (http://es.wikipedia.org/wiki/HAL_9000) is that the least Primitive Root for Wilson Primes is 2
I can't see any good reason for Wilson primes to have primitive root 2 with probability greater than the expected ~37% (= Artin's constant).

October 26th, 2009, 01:18 PM   #8
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Re: Primitive root of known Wilson Primes

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 I can't see any good reason for Wilson primes to have primitive root 2 with probability greater than the expected ~37% (= Artin's constant).
Thank you, now I know something new to me:

http://mathworld.wolfram.com/ArtinsConstant.html

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