## Re: primes between n^2 and (n+1)^2. Primes between n^3 and (n+1)^3.

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• In the same way one proves that the number of primes between cubes amounts to approximately: N = pi(n+1)^3-pi(n^3) ~ (n/ln(n))*(n+1) ~ pi(n)*(n+1), better ~
Message 1 of 4 , Apr 12, 2007
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In the same way one proves that the number of primes between
cubes amounts to approximately:

N = pi(n+1)^3-pi(n^3) ~
(n/ln(n))*(n+1) ~
pi(n)*(n+1),
better ~ n*pi(n).

WDS

--- In primenumbers@yahoogroups.com, "Werner D. Sand"
<Theo.3.1415@...> wrote:
>
> Proof:
>
> pi[(n+1)^2]-pi(n^2) ~ (PNT)
> [(n+1)^2]/ln[(n+1)^2] - (n^2)/ln(n^2) =
> [(n+1)^2]/2ln(n+1) - (n^2)/(2ln n) =
> [(n+1)^2]/2[ln n + ln(1+1/n)] - (n^2)/(2ln n) -> (n->inf)
> [(n+1)^2]/(2ln n) - (n^2)/(2ln n) =
> (2n+1)/(2ln n) =
> n/ln n + 1/(2ln n) -> (n->inf)
> n/ln n ~
> pi(n)
>
> qed
>
> Werner
>
>
> --- In primenumbers@yahoogroups.com, "Mark Underwood"
> <mark.underwood@> wrote:
> >
> > --- In primenumbers@yahoogroups.com, "Mark Underwood"
> > <mark.underwood@> wrote:
> > >
> > >
> > > As we know, the number of primes up to n is about n/log(n).
Given
> > > this, it is easy to show that the number of primes between n^2
and
> > > (n+1)^2 is also about n/log(n).
> > >
> > > How much does the actual count of primes between n^2 and (n+1)^2
> > > differ from n/log(n) ? On a very cursory inspection, it seems
the
> > > prime count is no more than sqrt(n) removed from n/log(n).
> > >
> > > Mark
> > >
> >
> >
> > Just did some prime counting and so far it holds that
> > the number of primes between n^2 and (n+1)^2 is within the range
> > n/log(n) +/- sqrt(n). At least for n up to about 47,000.
> >
> > There have been some close calls though. Here are the cases where
> the
> > difference between the actual prime count and n/log(n) was at
least
> 75
> > percent of the square root of n.
> >
> > Format: (n, primecount between n^2 and (n+1)^2, percentage)
> >
> > (696,81)
> > (696,85,81)
> > (1760,204,75)
> > (2456,268,94)
> > (2761,390,79)
> > (3516,486,93)
> > (3788,508,78)
> > (5266,675,83)
> > (9980,1168,84)
> > (10706,1250,93)
> > (15646,1718,78)
> > (23515,2458,79)
> > (23924,2503,84)
> > (28678,2923,76)
> > (28678,2923,76)
> > (32460,2986,77)
> > (39590,3904,83)
> >
> > Mark
> >
>
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