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