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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(p1)[,1]);for(n=2,p1,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 humanlike manner. I think you are a machine. 
October 23rd, 2009, 11:18 PM  #4  
Member Joined: Jan 2009 Posts: 95 Thanks: 0  Re: Primitive root of known Wilson Primes Quote:
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  
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 Quote:
(Love the .sig, by the way.)  
October 24th, 2009, 09:15 AM  #6  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Primitive root of known Wilson Primes Quote:
Quote:
 
October 26th, 2009, 10:59 AM  #7  
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 Quote:
 
October 26th, 2009, 01:18 PM  #8  
Member Joined: Jan 2009 Posts: 95 Thanks: 0  Re: Primitive root of known Wilson Primes Quote:
http://mathworld.wolfram.com/ArtinsConstant.html  

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