Finding prime numbers.
Goal: Given an integer N (N > 1), find all the prime numbers less than N.
The plan:
- A prime number is a number with no factors (integer divisors) other than itself and 1.
- We can check if X is divisible by Y using
X % Y == 0 - We can find all the numbers from 2 to N with a for-range loop.
- We can find only prime numbers by excluding non-primes from our result list.
- The only candidate divisors for X are between 2 and X, so if none of those evenly divide X then X is prime.
Version 1:
# ref: https://t5k.org/howmany.html
def is_prime(xx: int) -> bool:
for ii in range(2, xx):
if xx % ii == 0:
return False
return True
def list_primes(nn: int) -> list[int]:
"""List all the primes below nn."""
ys = []
for ii in range(2, nn):
if is_prime(ii):
ys.append(ii)
return ys
def count_primes(nn):
return len(list_primes(nn))
import sys
if __name__ == '__main__':
nn = int(sys.argv[1])
if nn <= 100:
print(list_primes(nn))
print("count =", count_primes(nn))
Improvements:
- We only need to check candidate divisors that are prime. If X is divisible by a non-prime Y, then it’s also divisible by all of Y’s prime factors.
- We only need to check candidate divisors from 2..sqrt(X), because if some Y > sqrt(X) divides X then X/Y must be an integer from 2..sqrt(X) that also divides X.
- We can skip evens > 2.
