cs2370 Notes: 17 Primes

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.