A bit of number theory with SymPy

SymPy contains many number-theory-related routines: obtaining prime numbers, integer decompositions, and much more. We will show a few examples here.

Getting ready

To display legends using LaTeX in matplotlib, you will need an installation of LaTeX on your computer (see this chapter's Introduction).

How to do it...

1. Let's import SymPy and the number theory package:

   In [1]: from sympy import *
           init_printing() 
   In [2]: import sympy.ntheory as nt

2. We can test whether a number is prime:

   In [3]: nt.isprime(2011)
   Out[3]: True

3. We can find the next prime after a given number:

   In [4]: nt.nextprime(2011)
   Out[4]: 2017

4. What is the 1000th prime number?

   In [5]: nt.prime(1000)
   Out[5]: 7919

5. How many primes less than 2011 are there?

   In [6]: nt.primepi(2011)
   Out[6]: 305

6. We can plot , the prime-counting function (the number of prime numbers less than or equal to some number x). The famous prime number theorem states that this function is asymptotically equivalent to x...