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April 19th, 2013, 11:22 AM   #31
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by CRGreathouse The limit does not exist. Probably $\liminf_{n\to\infty}t(n)=1$ and $\limsup_{n\to\infty}t(n)=+\infty.$
Not easy to prove.
t(n) is not even known unless n reach some limit.

How could you know that t(127) is equal to 3 at some stage t and could not increase to 4 or more at stage t+s ?

 April 19th, 2013, 11:36 AM #32 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets If it exist a prime number p < 127! but equal to (2k+1)p-1 where k>0 then t(127) will increase by one. So you will not even know the value of t(k) until you reach the value of k! minus something.
 April 19th, 2013, 04:50 PM #33 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets Here is an example of t(10) You do not have 10 in your list above of ks. But if we compute 10! mod 329891=329890 so t(10) will appear in the list and be equal to 1. 329891 is a certified prime (Wims) It is easy to find multiprimes starting from the factorial k. It will give you the definitive value of t(k) Im sure that I will find another increment of t(10).
 April 19th, 2013, 04:59 PM #34 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets I did the test until 10!=3628600. Idid not find any other value so Im sure that t(10)=1 (I hope I did not make mistakes).
April 19th, 2013, 06:34 PM   #35
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha t(n) is not even known unless n reach some limit.
I can find t(n) for small n. For large n the problem takes too much computational power.

Quote:
 Originally Posted by Mouhaha How could you know that t(127) is equal to 3 at some stage t and could not increase to 4 or more at stage t+s ?
t(127) = 8.

It's pretty clear that t(n) is finite. I could give you the proof but that would rob you of the ability to find it yourself (though if you insist...). I can prove that $\limsup_{n\to\infty}t(n)\ge2$ but I can't prove that this is infinite at the moment. The value of the lim inf is an open question for the time being.

April 19th, 2013, 06:37 PM   #36
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha I did the test until 10!=3628600. Idid not find any other value so Im sure that t(10)=1 (I hope I did not make mistakes).
t(10) = 2. There's a six-digit prime you missed.

April 19th, 2013, 07:07 PM   #37
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Mouhaha I did the test until 10!=3628600. Idid not find any other value so I`m sure that t(10)=1 (I hope I did not make mistakes).
t(10) = 2. There's a six-digit prime you missed.
Which one?

329891

I did not miss it. Read above.

329891 is a multiprime
11 is a uniprime.
It seems to me that here is misunderstandings.
See you tomorrow.

April 19th, 2013, 07:21 PM   #38
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Re: Splitting prime numbers in 2 sets

I don't see any such restriction in the definition of t:

Quote:
 Originally Posted by Mouhaha Let us call t(k) the quantity of primes giving p-1 as remains.

April 20th, 2013, 04:33 AM   #39
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
I don't see any such restriction in the definition of t:

Quote:
 Originally Posted by Mouhaha Let us call t(k) the quantity of primes giving p-1 as remains.
You are right.
But I have to redefine t(k) because for uniprimes k should be = 0. It makes more sense.
For the multiprimes k start from 1 to undefined value.
Now there are some results to prove or at least to understand.
The idea of splitting the set of prime numbers in 2 might lead to some unknown and maybe interesting territory or to somethng unuseful.
I do not know.

 April 20th, 2013, 06:50 AM #40 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: Splitting prime numbers in 2 sets Well, I suspect that the limit does not exist for this new function either, and that the limit inferior and limit superior are just like the original t(n).

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