November 24th, 2007, 06:28 PM  #1 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  prime frequency
will there always be a prime between a given prime and the least prime greater than the square root of the first prime?

November 24th, 2007, 06:56 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 
Yes, for p >= 7. By Bertrand's postulate we know that there is a prime strictly between n/2 and n for n > 2, and sqrt(n) <= n/2 for n >= 4. So there is a prime strictly between sqrt(p) and p for all p >= 4. Since you want at least two primes (since you want one strictly between nextprime(sqrt(p)) and p), you'll need the extended form of Bertrand's postulate, proved as I recall by Ramanujan, that shows that for large enough n there are at least k primes between n and 2n. Edit: Here's Ramanujan's paper: see equation (18) at the bottom. 
November 24th, 2007, 08:02 PM  #3 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  not what i meant
what i meant was p<p_1<p+(sqrt(p) rounded up the the nearest prime) 
November 24th, 2007, 08:09 PM  #4 
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 
In that case, you're essentially asking about Legendre's Conjecture or some variant on it. I don't think there's any hope for a quick solution. Even Schoenfeld's version of the Riemann hypothesis seems too weak to prove this.

November 24th, 2007, 10:23 PM  #5 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  not sure
but i dont think so, I am doing this: p_0<P_1<P_0+(Sqrt(p) rounded up to the nearest prime). how is that the same as: there is a prime between p^2 and (p+1)^2? 
November 25th, 2007, 03:18 AM  #6  
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Re: not sure Quote:
p_0+sqrt(p_0)= 23+4 (or 5) =27 or 28 p_1=29 ???  
November 25th, 2007, 11:01 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: not sure Quote:
The generalized Legendre conjecture is that there are some integers K, N where there is a prime between n and K sqrt(n) for all n > N. Legendre predicted in particular that it would hold for some N (probably 1) and K = 2. You're asking if it holds for K = 1.  
November 25th, 2007, 06:29 PM  #8 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  not really
there is a difference as i am doing this: between n and n+(sqrt(n) rounded up to the nearest prime). not between n and k(sqrt(n)) 
November 25th, 2007, 06:56 PM  #9  
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: not really Quote:
 
November 26th, 2007, 04:52 AM  #10 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  no
i start out with a given p_0 2003 for example i then take the square root ~44.75 i then round that up to the nearest prime 47 then i say that there is a prime between 2003 and 2003+47 there are: 2011 2017 2027 2029 2039 does make it clearer? 

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