Re: Percentage of Prime Numbers

On 11 May, 13:12, Narcoleptic Insomniac
<i_have_narcoleptic_insom...@xxxxxxxxx> wrote:
On May 11, 2008 6:36 AM CT, ad...@xxxxxxxxxxxx wrot:

Hi,

I just checked my 10-€-Banknote and found out the
serial number is a prime number. I just asked the
European Central Bank how many Banknotes circulate and
want now to find out the chance that a randomly picked
banknote has a prime serial number. So - is there any
possibility (without crashing my PC) to find out how
many prime numbers are in a given area of numbers?

Thanks a lot!

Yes, one way would be to use approximations of the prime
counting function (denoted pi(x)).  The Prime Number
Theorem tells us that this function is asymptotic to
x / ln(x), or in other words

pi(x) ~ x / ln(x).

So, suppose that you wanted to know roughly how many
primes were between A and B (given B > A).  Then, by the
PNT, we know that

pi(B) - pi(A) ~ B / ln(B) - A / ln(A)

= B ln(A) - A ln(B) / [ln(B) ln(A)]

= ln(A^B) - ln(B^A) / [ln(B) ln(A)]

= ln(A^B / B^A) / [ln(B) ln(A)]

..is roughly the number of primes between A and B.

Of course, you can divide this by (B - A) to get an
approximate density.

Regards,
Kyle Czarnecki


that is cool. any idea why?
.

Relevant Pages

  • Re: Percentage of Prime Numbers
    ... On May 11, 2008 6:36 AM CT, admin@xxxxxxxxxxxx wrot: ... European Central Bank how many Banknotes circulate and ... counting function ). ... approximate density. ...
    (sci.math)
  • Re: Percentage of Prime Numbers
    ... I just asked the European Central Bank ... Banknotes circulate and want now to find out the ... possibility (without crashing my PC) to find out how ... An intriguing and interesting question. ...
    (sci.math)
  • Re: Percentage of Prime Numbers
    ... I just asked the European Central Bank how many ... Banknotes circulate and want now to find out the chance that a ... idea, however, how the Central Bank assigns serial numbers. ... your own notes substantially harder. ...
    (sci.math)
  • Percentage of Prime Numbers
    ... I just asked the European Central Bank how many ... Banknotes circulate and want now to find out the chance that a ... randomly picked banknote has a prime serial number. ... possibility (without crashing my PC) to find out how many prime ...
    (sci.math)