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

[Richard Guy]

Fools rush in ...

I believe that it would follow from the
Riemann hypothesis.     R.

On Tue, 12 Dec 2006, Dean Hickerson wrote:

> 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
>