I decided to take on Project Euler’s problem #10. Its statement goes like this:
The sum of the primes below 10 is [pmath]2 + 3 + 5 + 7 = 17[/pmath].
Find the sum of all the primes below two million.
My first attempt used brute force and testing primality using only previously found primes:
primes =  def is_prime(n): if not (n < 2 or any(n % x == 0 for x in filter(lambda x: x < math.ceil(n ** 2), primes))): primes.append(n) return True return False sum_primes = 0 n = 1 while n < 2000000: if (is_prime(n)): sum_primes += n n += 1 print sum_primes
It worked if you consider an execution time of over 7 minutes reasonable. Clearly a bad solution. So I decided to rethink it a bit and tried again. This time, I decided to use a little creative thinking. For every number I tested for primality, I took the time to mark its multiples as not primes (when I first test, say, 3, I already know that [pmath]6, 9, 12, … n * 3 < 2000000[/pmath] will not be a prime numbers, so I don’t have to test their primality later.)
max_primes = 2000000 numbers =  * max_primes def is_prime(n): if n <= 2: return True if numbers[n] == 1: return False c = 2 m = 0 while True: m = n * c if m < max_primes: numbers[m] = 1 else: break c+= 1 #if any(n % x == 0 for x in xrange(2, int(n ** 0.5) + 1)): i = 2 while i < int(n** 0.5 + 1): if n % i == 0: return False i += 1 numbers[n] = 1 return True def sum_primes(): soma = 0 for i in range (2, max_primes): if is_prime(i): soma += i return soma print sum_primes()
This is a much better algorithm and it gave the correct result in 54 seconds. Project Euler has a “one-minute rule”, which state that all its problems should be solvable “on a modestly powered computer in less than one minute,” which means the solution applies. Still, it’s a brute force solution and I wanted a better one.
So today I picked my copy of Concrete Mathematics and read about primes. Honestly, I’ve been reading more about primes lately than I even thought I would.
Anyways, Knuth—yes, I know the book has three authors, but I can’t help thinking of it as a Knuth book—talks about several strategies to test primality, but it also mentions sieve algorithms that are used to generate lists of prime numbers. Exactly what I wanted!
def prime_list(limit): if limit < 2: return  sieve_size = limit / 2 sieve = [True] * sieve_size for i in range(0, int(limit ** 0.5) / 2): if not sieve[i]: continue for j in range((i * (i + 3) * 2) + 3, sieve_size, (i * 2) + 3): sieve[j] = False primes =  primes.extend([(i * 2) + 3 for i in range(0, sieve_size) if sieve[i]]) return primes print reduce(lambda x,y: x+y, prime_list(2000000))
Et voilà! Correct result in 0.456s! The code is based on the description of the Sieve of Eratosthenes found in Concrete Mathematics. Also interesting to note, my idea in the second algorithm above is actually the basis of this sieve algorithm, I was just thinking “backwards.”