## Visualising the distribution of primes

Goal:

Prime numbers appear to be sporadic and hard to predict. If, however, one counts the number of primes less than x, then one can prove (that's the famous prime number theorem) that this number is asymptotically equal to x/log(x). This means that the n-th prime, call it p_n, is asymptotically of size n log(n), i.e. the distance between primes gets bigger and bigger (on average).

The goal is to visualise the distribution of primes. One can do this by

• plotting the prime counting function and comparing it with x/log(x),
• plotting the integer nearest to p_n/log(n),
• plotting the Klauber triangle or the Ulam spiral (see e.g. on wikipedia),
• other ways that you may devise.

This subject is related to the project on the testing conjectures on primes.

Participants: Jim Barthel, Pietro Sgobba, Fa Zhu.

Supervisors: Gabor Wiese, Panagiotis Tsaknias.

Difficulty level: Experimental Mathematics 3

Tools: Computer Algebra system like SAGE.

Results: Project report