Re: The nth Prime



On May 16, 10:59 am, "Dirk Van de moortel" <dirkvandemoor...@ThankS-NO-
SperM.hotmail.com> wrote:
bill <b92...@xxxxxxxxx> wrote in message

53f02011-eca4-41b2-8059-a1e7630e3...@xxxxxxxxxxxxxxxxxxxxxxxxxxx



P(n) is the nth prime. There are n primes less than or equal to
P(n).

Therefore, n = pi [ P(n) ] ~ P(n) / ln[ P(n) ] Rearranging;

P(n) ~ n* ln[ P(n) ]

If X = n * ln(n), then

P(n) ~ n * ln(X)

For n's up to 1,000,121,668,853 this formula gives consistently
better estimates for P(n) than "n * ln(n)". For the same range; it
also
gives better estimates than "n * ln[ P(n)]".

It is possible that as 'n' increases; that "n * ln(n)" or "n *
ln[ P(n)]" may
give a more accurate estimate of P(n).

If we write out the expression we get;

P(n) ~ n * ln[ n* ln(n)] Is it possible to simplify the right hand
term?

Bill J

Of course n ln( n ln(n) ) = n ln(n) + n ln( ln(n) ).
But then you should worry about simplifying the next better approximation:
http://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_...
;-)

Dirk Vdm

Thanks - for nothing. I thought that I had discovered
something new.

Bill J
.