Friday, January 30, 2009

The Sequence of Primes

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 pn and pi(n) and other aspects of the sequence that manifest themselves in the first couple of thousand terms.

Here is one of the places you can get your list of primes: http://primes.utm.edu/. 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 = 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 nth 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 was made by bringing the same data into Tinkerplots - it suggests the distribution of primes and twin primes (in blue) in the numbers up to the 2262nd prime.