New sequence : Primes for which SQRT(A000040(n)) < A001223(n)

[David Wilson]

My intuition was that lim n->inf (p(n)/log(p(n))) / n = 1 would preclude an 
infinitude of gaps p(n+1)-p(n) >= sqrt(n). Can someone give a function p 
that would serve as a counterexample? If so, I will give my intuition a 
sound boxing to the ears.

----- Original Message ----- 
From: "Dean Hickerson" <dean at math.ucdavis.edu>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, December 12, 2006 11:49 AM
Subject: Re: New sequence : Primes for which SQRT(A000040(n)) < A001223(n)


> Mostly to David Wilson:
>
> Concerning primes p for which there are no primes between p and p+sqrt(p),
> you asked:
>
>> Wouldn't the finiteness of this sequence follow from the prime number
>> theorem?
>
> Not that I can see.  Do you have a proof?
>
> Note that finiteness of this sequence implies that, for every sufficiently
> large positive integer n, there's a prime between n^2 and (n+1)^2.  Except
> for the "sufficiently large", that's Legendre's conjecture, which is still
> unproved.
>
> Dean Hickerson
> dean at math.ucdavis.edu