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# Prime Counter

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)

Explore the percentage accuracy of this approximation.

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)

$$

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Age 16 to 18

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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)

Explore the percentage accuracy of this approximation.

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.

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.

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