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

> >

>