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) ?) * * * 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). --Dan
--- dasimov@earthlink.net wrote: ...
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).
--Dan
In Harvey Cohn, "Advanced Number Theory" a proof is given that the primes in arithmetic progression are equidistributed among residue classes in the sense that, for gcd(a,m) = 1, sum(1/p^s, p == a mod m) 1 ------------------------ ==> ------ as s ==> 1. sum(1/p^s, all primes p) phi(m) Cohn states without proof, and without citation, that as x ==> infinity, Number of primes < x that are == a mod m 1 ---------------------------------------- ==> ------ Number of primes < x phi(m) is difficult but true. Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
----- 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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