As I make my way through Hardy & Wright’s An Introduction to the Theory of Numbers, I am hoping to work it into my recreational math pursuits – coming up with interesting (but not too heavy) activities that correspond roughly to the material in the text.

The first two chapters are on the sequence of primes. Here’s the activity: obtain a list of primes, import them into Fathom, and construct plots that explore *p*n and *pi*(*n*) and other aspects of the sequence that manifest themselves in the first couple of thousand terms.

In my Fathom experiment, I imported the first 2262 prime numbers.

If you import a sequential list of primes into Fathom (under the attribute **prime**) and add another attribute** n=caseindex**, you can create two nice plots. Plot A should have **prime **as the *x* axis and **n** as the *y* axis. This shows the function *pi*(*n*). To this plot you should add the function *y *= *x*/*ln*(*x*) and visually compare the two curves. Plot B should have the *x* and *y* axis reversed. On this graph, plotting the function *y* = *x***ln*(*x*) shows how closely this approximation for *pn* (the *n*th prime) comes to the actual values.

You can add further attributes to look at the distance between primes **dist=prime-prev(prime)**, and also the frequency of twin primes **is_twin = (dist=2)or(next(dist)=2)**.

You can also add attributes to keep a running count of **twin_primes**, and to keep a running average of the **twin_primes**. The plot above shows how the ratio of tiwn primes diminishes as the number of primes increases. The plot at the top of the post suggests the distribution of primes and twin primes (in blue) in the numbers up to the 2262nd prime.

For more such insights, log into www.international-maths-challenge.com.

Credit for article given to dan.mackinnon*****