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April 17th, 2013, 10:28 AM   #1
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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 "uni-prime" if not it will then be called "multi-prime".

Example :
17 is uni-prime
23 is multi-prime
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April 17th, 2013, 11:38 AM   #2
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by Mouhaha
Is there another mathematical procedure to split the prime numbers set in 2 sets (only).
The number of ways to partition the primes into two sets is uncountably infinite. The number of definable ("mathematical"?) ways is countably infinite.

I can't tell if your example is standard or not since it lacks a definition and has only 2 examples.
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April 17th, 2013, 11:46 AM   #3
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Re: Splitting prime numbers in 2 sets

Sorry I forget to send the definition of multi-prime.
If (p-1)! mod p = p-1 then p is prime (Wilson theorem).
If we found AT LEAST one number k <p-1 such as k! mod p = p-1 then the number p is called multi-prime, otherwise it is called uni-prime.

Example : 23 is multi-prime 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 uni-primes when n goes infinite
- The limit of the inverses of the set of multi-primes when n goes infinite
- The limit of Mertens-like function (let us call it Mouhaha function ) where :
Mouhaha(uniprime)=-1
Mouhaha(multiprime)=+1
Mouhaha(composite)=0
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April 17th, 2013, 01:19 PM   #4
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by Mouhaha
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 "uni-prime" if not it will then be called "multi-prime".

Example :
17 is uni-prime
23 is multi-prime
17 is 4n + 1. 23 is 4n + 3. There's a big difference between these two classes of prime numbers. As one example, these two classes are the two different cases of Gauss's law of quadratic reciprocity.

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.
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April 17th, 2013, 02:05 PM   #5
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Re: Splitting prime numbers in 2 sets

What you call a multi-prime 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 "Mertens-like 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.
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April 17th, 2013, 02:18 PM   #6
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
What you call a multi-prime 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 "Mertens-like 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.
The first 2000 will not give on how a Mertens like function will behave as n goes infinite.
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
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April 17th, 2013, 02:43 PM   #7
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by Maschke
Quote:
Originally Posted by Mouhaha
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 "uni-prime" if not it will then be called "multi-prime".

Example :
17 is uni-prime
23 is multi-prime
17 is 4n + 1. 23 is 4n + 3. There's a big difference between these two classes of prime numbers. As one example, these two classes are the two different cases of Gauss's law of quadratic reciprocity.

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.
23 form 4n+3 is multiprime
but 29 form 4n+1 is multiprime too
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April 17th, 2013, 03:23 PM   #8
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by Mouhaha
The first 2000 will not give on how a Mertens like function will behave as n goes infinite.
I gave a heuristic which predicted a particular behavior. I only showed the first 2000 for comparison with that prediction.

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:
Originally Posted by Mouhaha
My feeling is that it is too premature to claim this or that.
It needs lot of work before reaching conclusion.
Do you have a suggestion for a different heuristic, or a tactic toward an actual proof?
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April 17th, 2013, 04:18 PM   #9
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Re: Splitting prime numbers in 2 sets

I take it that 2 is a multiprime since it divides 0!+1?
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April 17th, 2013, 04:22 PM   #10
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by johnr
I take it that 2 is a multiprime since it divides 0!+1?
Yes. Its status does not affect any of the questions we're working on, though.
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