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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? Tags frequency, prime Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post kucinglover Advanced Statistics 1 October 22nd, 2012 12:42 PM dkellerm Physics 1 February 2nd, 2012 02:55 PM skarz Advanced Statistics 1 February 3rd, 2010 08:40 AM armandine2 Applied Math 0 July 19th, 2009 05:17 AM dkellerm Algebra 0 December 31st, 1969 04:00 PM

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