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

[Dean Hickerson]

Remi Eismann wrote:

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

No.  Cramer's conjecture says that for any positive number C<1, there are
infinitely many primes p for which there are no primes between p and
p + C log(p)^2,  and for any C>1, there are only finitely many such primes p.
It doesn't say anything about the case C=1.

> Primes p for which there are no primes between p and p+sqrt(p) :
> 3,7,13,23,31,113
> srqt(p) > ln(p)^2 for p=2, p=3 and p in (5507, infinity)
> this sequence is finite and complete because of the Cramer's conjecture.

Cramer's conjecture would imply that the sequence is finite, but not that
it's complete.

> Cramer made his conjecture considering the Riemann hypothesis as true.

I don't think he made that assumption.


 From what I've read, Cramer's conjecture is based on a probabilistic model
of primes:  If you define a set S by independently putting each positive
integer n into S with probability 1/log(n), then, with probability 1,
S will satisfy  lim sup (s(n+1)-s(n))/log(s(n))^2 = 1,  where s(n) is the
i'th element of S.  (So assuming that the set of primes is typical among
such sets, it's reasonable to guess that Cramer's conjecture is true.)
Does anyone on this list know how that result is proved?  And what is the
probability that such a set satisfies  s(n+1)-s(n) < log(s(n))^2  for
all n?  Or for all sufficiently large n?

Dean Hickerson
dean at math.ucdavis.edu