Re: [math-fun] a prime in each row?
David Wilson writes: << Propp's conjecture* would imply a prime between n^2 and (n+1)^2, which conjecture I believe is stil outstanding.
*That for any n >=2 and for each k in [0,n-1], there is at least one prime in the "row" of n consecutive numbers [kn+1,(k+1)n]. ------------------------------------------------------------------------------ ---------------------------------- I guess David Wilson's point is that the interval of 2n numbers [(n-1)^2, n^2], forms the last two rows in Propp's conjecture. But obviously, there's a prime in the first row, and there's one in the second row guaranteed by Bertrand's Postulate. Question: Where does the conjectural territory first begin? [2n+1,3n] ? [3n+1,4n] ? Etc. --Dan
asimovd@aol.com wrote:
But obviously, there's a prime in the first row, and there's one in the second row guaranteed by Bertrand's Postulate.
Question: Where does the conjectural territory first begin? [2n+1,3n] ? [3n+1,4n] ? Etc.
It follows from the prime number theorem that, given e>0, there exists N=N(e) such that for all x > N, there is a prime between x and (1+e)x. Thus for n large enough there is a prime between 2n+1 and 3n, and also 3n+1 and 4n, and in fact any kn+1 and (k+1)n for fixed k independent of n. Gary McGuire
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
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asimovd＠aol.com -
Gary McGuire -
John Conway