Copyright © University of Cambridge. All rights reserved.

'Weekly Challenge 44: Prime Counter' printed from http://nrich.maths.org/

Show menu




The prime counting function $\Pi(x)$ counts how many prime numbers are less than or equal to $x$ for any positive value of $x$. Since the primes start $2, 3, 5, 7, 11, 13, \dots$ we therefore have, for example, $\Pi(11) = 5$ and $\Pi(8) = 4$.

It is believed by mathematicians that $\frac{x}{\ln(x)}$ is a good approximation to $\Pi(x)$. It is believed to get progressively better as $x$ increases to very, very large numbers. How well does it work for lower values of $x$ (up to the 100,000th prime)

Use the following interactivity to examine the percentage accuracy of this approximation for these values.


If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.


Use a few sensible values / choice of axes to try to create a useful graphical representation of $\ln(\Pi(x))$ against $\ln(x)$ for $x$ taking values up to about a million. Use your curve to try to predict $\Pi(x)$ for a few values of $x$ away from your data points. How close are your estimates for whole number multiples of $100,000$?

Use your judgement to try to extrapolate your curve to make approximations as to
$$
\Pi(10^7)\quad\quad \Pi(10^8)\quad\quad \Pi(10^9)
$$



Did you know ... ?

Although prime numbers are distributed with some large-scale regularity across the natural numbers, as this problem indicates, there is no known process which generates the prime numbers.