Python number theory

Whole-number tricks: n % d == 0 means d divides n; test divisors only up to the square root; the sieve of Eratosthenes finds every prime up to n at once.

pow(a, b, m) works out a to the power b, modulo m, quickly, even when the power is huge.

n = 50
is_prime = [True] * (n + 1)
is_prime[0] = is_prime[1] = False
for i in range(2, int(n ** 0.5) + 1):
    if is_prime[i]:
        for j in range(i * i, n + 1, i):
            is_prime[j] = False
print(sum(is_prime))

Watch for: Testing every number up to n one by one is too slow for big inputs: the hidden tests notice.