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 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
 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
November 25th, 2007, 03:18 AM   #6
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Re: not sure

Quote:
 Originally Posted by soandos but i dont think so, I am doing this: p_0
p_0=23

p_0+sqrt(p_0)= 23+4 (or 5) =27 or 28

p_1=29

???

November 25th, 2007, 11:01 AM   #7
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Re: not sure

Quote:
 Originally Posted by soandos I am doing this: p_0
(p+1)^2 = p^2 + 2p + 1 = p^2 + 2sqrt(p^2) + 1 ~= p^2 + 2sqrt(p^2)

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
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Re: not really

Quote:
 Originally Posted by soandos 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))
Not really, it just means you're looking for two primes instead of one.

 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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