----- Original Message ----- From: <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, July 04, 2006 11:32 AM Subject: Re: [math-fun] Factorial n
David Wilson wrote:
<< 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.
Does "for sufficient n" mean for n >= N(r,m,e), so that possibly N(r,m,e) -> oo as e -> 0 ? (Or might it be just for n >= N(r,m) ?)
Yes, it is N(r,m,e) with N(r,m,e) -> oo as e -> +0.
I would be willing to conjecture that for integers m > r > 0, GCD(r,m) = 1, then the fraction of primes == r (mod m) is asymptotically r/m.
Does anyone know a counterexample to that? It seems that it might be a consequence of David's conjecture (or perhaps vice versa).
According to a recent post, this was proved by Vallee-Poussin. It was really a pretty safe conjecture on my part, as I seem to remember reading material on the distribution of primes == k (mod n).
--Dan
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