Project Euler #10: Summation of primes

This post is copied from another blog I run. I'll be shutting that blog down shortly and all future posts of that kind will go on this blog. These are solutions to various programming brain teasers. Optimizing your solution is the real challenge.

The problem reads as follows:

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

Right from the start I can think of two methods:

1. Using a similar solution as for PE-7 but adding the primes instead of storing them
2. Sieve of Eratosthenes

I opted for the first method since I'm lazy. With minimal modifications, I get 173518 microseconds as my runtime. I was curious about the second method, so I copied and modified a solution found in the problem discussion thread. You can find that one under sieve.cpp. It has a runtime of 221631 microseconds. That's still pretty good.

The third way I can think of is to combine the first two methods. As with PE-7, automatically exclude all multiples of 2 and 3 for a 2/3 reduction in numbers considered. Then find all the primes below the square root of 2 million (so anything below and including 1415) and sieve on the remaining 1/3 with those prime numbers. Could result in a nice speedup over the first method. As usual, another thing you could do is to remove the use of the modulus operand - it's expensive.

The code can be found here