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

[reismann at free.fr]

Dear Richard, Dean, David and Seqfans

Primes p for which there are no primes between p and p+ln²(p) :
2,3,7
Conjecture : this sequence is finite and complete = Cramer's conjecture (If I
understood well)

Primes p for which there are no primes between p and p+sqrt(p) :
3,7,13,23,31,113
srqt(p) > ln² p for p=2, p=3 and p in (5507, infinity)
this sequence is finite and complete because of the Cramer's conjecture.
Cramer made his conjecture considering the Riemann hypothesis as true.
Is it true ?

Best

Rémi EISMANN

Selon Richard Guy <rkg at cpsc.ucalgary.ca>:

> 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
> >
>