### Weekly Challenge 43: A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

### Weekly Challenge 28: the Right Volume

Can you rotate a curve to make a volume of 1?

### Weekly Challenge 46: the Jabber-notty

Can you invert this confusing sentence from Lewis Carrol?

# Weekly Challenge 44: Prime Counter

##### Stage: 5 Challenge Level:

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.

This text is usually replaced by the Flash movie.

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.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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