since nesting n+n^(1/2) on 1 produces A002984,(a(n) = a(n-1)+[sqrt a(n-1)]) and changing floor to ceiling, we get A002620 (a(n) = Floor[n^2 /4] ), we should count primes in intervals from Floor[n^2 /4] to Floor[(n+1)^2 /4] giving, I recon : {0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 4, 3, 1, 4, 2, 4, 3, 3, 3, 4, 4, 3, 4, 3, 2, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 4, 4, 4, 5, 5, 5, 4, 6, 4, 4, 5, 5, 5, 7, 2, 3, 6, 6, 6, 6, 5, 8, 4, 5, 6, 5, 4, 7, 5, 4, 7, 6, 7, 7, 3, 6, 7, ... (not yet EIS) de amatoribus nihil sed bonum. W. -----Original Message----- From: John Conway [mailto:conway@Math.Princeton.EDU] Sent: donderdag 5 juni 2003 17:07 To: math-fun Subject: Re: [math-fun] a prime in each row? On Thu, 5 Jun 2003 asimovd@aol.com wrote:
David Wilson writes:
<< Propp's conjecture* would imply a prime between n^2 and (n+1)^2, which conjecture I believe is stil outstanding.
Question: Where does the conjectural territory first begin? [2n+1,3n] ? [3n+1,4n] ? Etc.
--Dan
All of these are fine - we know for any positive epsilon that there's a prime between n and n(1+epsilon) for all sufficiently large n (and explicit bounds can be given for how large n need be). The right question to ask is for which c there's necessarily a prime between n and n + n^c (for all sufficiently large n). The Riemann hypothesis would tell us that this is true for any c > 1/2. Ingham proved it for c = 3/5, remarking that his methods would give a strictly smaller number, and later people have explicitly produced values that are strictly smaller, but not by very much. John conway _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.