April 17th, 2013, 11:28 AM  #1 
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Splitting prime numbers in 2 sets
Hi, Is there another mathematical procedure to split the prime numbers set in 2 sets (only). It is obvious that we can choose any arbitrary criterion to do it such the primes finishing by (2,3,5,7) and (1,9) or anything like that. I have found a way to split the set of prime numbers using Wilson theorem. If a prime number fit some condition then it will be called "uniprime" if not it will then be called "multiprime". Example : 17 is uniprime 23 is multiprime 
April 17th, 2013, 12:38 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Splitting prime numbers in 2 sets Quote:
I can't tell if your example is standard or not since it lacks a definition and has only 2 examples.  
April 17th, 2013, 12:46 PM  #3 
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Splitting prime numbers in 2 sets
Sorry I forget to send the definition of multiprime. If (p1)! mod p = p1 then p is prime (Wilson theorem). If we found AT LEAST one number k <p1 such as k! mod p = p1 then the number p is called multiprime, otherwise it is called uniprime. Example : 23 is multiprime because it exists a number k=18 < 22 such as 18! mod 23=22 I do not know if THE SPLITTING in 2 sets will lead to interesting consequences. What is interesting is to compute 3 things :  The limit of the inverses of the set of uniprimes when n goes infinite  The limit of the inverses of the set of multiprimes when n goes infinite  The limit of Mertenslike function (let us call it Mouhaha function ) where : Mouhaha(uniprime)=1 Mouhaha(multiprime)=+1 Mouhaha(composite)=0 
April 17th, 2013, 02:19 PM  #4  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Re: Splitting prime numbers in 2 sets Quote:
http://en.wikipedia.org/wiki/Quadratic_reciprocity If you're interested in the relation to Wilson's theorem, you might want to read this ... http://www.math.ou.edu/~kmartin/nti/chap6.pdf It's not special, I just found it by a random Google search. Lots of other info out there on the distinctions between the 4n+1 and the 4n+3 primes.  
April 17th, 2013, 03:05 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Splitting prime numbers in 2 sets
What you call a multiprime appears as Sloane's A166862. Unfortunately the questions you ask about it are not answered there. A naive heuristic would treat k! mod p as independent random variables and conclude that the fraction of primes which are uniprimes is 1/e and the fraction which are multiprimes is 1  1/e. That would mean that the limit of your "Mertenslike function" would be as would the limit of the inverses of both sets. As a quick check, I tested the first 2000 primes and found 1272 multiprimes. The heuristic suggests that I would find about 1264. 
April 17th, 2013, 03:18 PM  #6  
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Splitting prime numbers in 2 sets Quote:
My feeling is that it is too premature to claim this or that. It needs lot of work before reaching conclusion. Ps : I`m trying to understand the difference between the 2 sets. There is maybe some quick computation to know to which a set belong some chosen prime. Thanx  
April 17th, 2013, 03:43 PM  #7  
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Splitting prime numbers in 2 sets Quote:
but 29 form 4n+1 is multiprime too  
April 17th, 2013, 04:23 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Splitting prime numbers in 2 sets Quote:
It may be of interest that in the first 16,000 primes there are 10,305 multiprimes, which fits the prediction of 10,113 pretty closely as well. Feel free to test further if you like. (The expected variance is 3720 so it's not quite 3 sigma, but it's not surprising to see a bias in that direction since the small numbers are always excluded.) Quote:
 
April 17th, 2013, 05:18 PM  #9 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Splitting prime numbers in 2 sets
I take it that 2 is a multiprime since it divides 0!+1?

April 17th, 2013, 05:22 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Splitting prime numbers in 2 sets Quote:
 

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