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

[reismann at free.fr]

Hi,

Thank you very much for these explanations. But I am not sure to have understood
everything. I am going to continue to work...

A Granville's study about Cramér which seems complete :
http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

In january, I will submit two sequences :
1) Primes p for which there are no primes between p and p+sqrt(p) :
3,7,13,23,31,113
with the conjecture : this sequence is finite and complete
Comments are welcomes.

2) Primes p smaller than sqrt(g)*exp(sqrt(g)), with g gap of p :
2, 3, 5, 7, 13, 23, 113, 1327, 31397, 370261, 2010733, 20831323...
Comments and additional terms are welcomes.

I can't submit this sequence, there are too few terms :
Primes p for which there are no primes between p and p+ln(p)^2 :
2,3,7

Thank you all very much for this discussion which obliges me to improve myself
(math and english).
Have a nice day

Rémi Eismann
Selon Dean Hickerson <dean at math.ucdavis.edu>:

> 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
>