Problem: The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17. Find the sum of all the primes below two million.

### Sage Solution

There is a direct Sage solution here. We’ll create a list of all primes under two million and sum over that list. We’ll also throw a time component in front so we know how long Sage takes to do the computations.

time sum(prime_range(2e6))

Executing this gives the following result.

142913828922 Time: CPU 0.04 s, Wall: 0.05 s

### Python Solution

We’ll use our prime sieve from Project Euler Problem 7 and simply rewrite things a bit.

import time   def prime_sum(n): if n < 2: return 0 if n == 2: return 2 if n % 2 == 0: n += 1 primes = [True] * n primes[0],primes[1] = [None] * 2 sum = 0 for ind,val in enumerate(primes): if val is True and ind > n ** 0.5 + 1: sum += ind elif val is True: primes[ind*2::ind] = [False] * (((n - 1)//ind) - 1) sum += ind return sum   start = time.time() sum = prime_sum(2000000) elapsed = (time.time() - start)   print "found %s in %s seconds" % (sum,elapsed)

This executes in under a second:

found 142913828922 in 0.838065862656 seconds

### Cython Solution

What is a bit amusing is that we can do a trivial rewriting to Cython and achieve a speedup.

%cython   import time   cdef prime_sum(unsigned long n): if n < 2: return 0 if n == 2: return 2 if n % 2 == 0: n -= 1 primes = [True] * n primes[0],primes[1] = [None] * 2 sum = 0 for ind,val in enumerate(primes): if val is True and ind > n ** 0.5 + 1: sum += ind elif val is True: primes[ind*2::ind] = [False] * (((n - 1)//ind) - 1) sum += ind return sum   start = time.time() sum = prime_sum(2000000) elapsed = (time.time() - start)   print "found %s in %s seconds" % (sum,elapsed)

When executed, we get:

found 142913828922 in 0.277922153473 seconds

The Cython code (which has minimal modifications) is about 3 times as fast as the Python code. Let’s see how much we can improve the Cython version. We’ll rewrite the list of integers as a pointer-based C array of Boolean values. This should speed things up a bit.

%cython   import time from libc.stdlib cimport malloc, free   # a function to sum all prime numbers below a given number cdef prime_sum(unsigned long n):   if n < 2: return 0 if n == 2: return 2 # if n is even, use next lowest number if n % 2 == 0: n -= 1   cdef bint *is_prime = <bint *>malloc(n * sizeof(bint))   # set all to prime and then sieve cdef unsigned long int i,j,sum = 0 i = 0 while i < n: is_prime[i] = 1 i += 1 #for i in range(n): is_prime[i] = 1 is_prime[0],is_prime[1] = 0,0 i = 2 while i < n ** 0.5 + 1: if is_prime[i] == 1: # add this to the sum sum += i # remove all multiples of i from prime list j = i ** 2 while j <= n: is_prime[j] = 0 j += i i += 1 while i <= n: if is_prime[i]: sum += i i += 1 free(is_prime)   return sum   start = time.time() sum = prime_sum(2000000) elapsed = (time.time() - start)   print "found %s in %s seconds" % (sum,elapsed)

When executed, we now get the following.

found 142913828922 in 0.0436608791351 seconds

The Cython executes roughly 19 times faster than the original Python code.