Michael Kleber already told us that Erdos had already proved my assertion that primes of the form 4n+1 and 4n+3 exist between n and 2n for n >= 7. I would be willing to conjecture that if there are an infinitude of primes == r (mod m), then there is a prime of this form between n and n(1+e) for sufficient n. Are there any results on the density of such primes relative to all of the primes? If so, then probably the prime number theorem could probably be used to establish the above conjecture. ----- Original Message ----- From: "Eric Bach" <bach@cs.wisc.edu> To: "David Wilson" <davidwwilson@comcast.net> Sent: Tuesday, July 04, 2006 1:53 AM Subject: Re: [math-fun] Factorial n
On Mon, 3 Jul 2006, David Wilson wrote:
No, I don't have a proof, I do have an argument following from a solid conjecture.
We know that for n >= 2, there exists a prime p with n < p < 2n. It also appears that for n >= 4, there exists prime p == 3 (mod 4) with n < p < 2n. I cannot prove this myself, but I am sure it is true.
Kevin McCurley had a paper (Math. Comp.?) giving explicit prime number theorems for arithmetic progressions. Probably your supposition would follow from his work plus some computer checking of small cases.
Eric
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