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October 23rd, 2009, 12:30 PM   #1
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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)
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October 23rd, 2009, 08:57 PM   #2
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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
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October 23rd, 2009, 10:48 PM   #3
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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.
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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
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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.)
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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...
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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).
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October 26th, 2009, 01:18 PM   #8
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Re: Primitive root of known Wilson Primes

Quote:
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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