素数检查程序是否不能消除5的整数？

Question

My program looks at an input and then return a list of all prime numbers less than or equal to the number. It currently is not eliminating numbers divisible by 5, and my 'Testing divisibility by' statement is returning 3 each time.

import math


def is_prime(number):
    if number % 2:
        # equivalent to if number % 2 != 0 because if number is
        # divisible by 2 it will return 0, evaluating as 'False'.
        for num in range(3, int(math.sqrt(number)) + 1, 2):
            print("Testing divisibility by {}".format(num))
            if number % num == 0:
               return False
            else:
                return True
    else:
        return False



def list_of_primes(number):
    prime_list = []
    for x in range(2, number + 1):
            if is_prime(x):
                prime_list.append(x)
    return prime_list



def main():
    while True:
        usr_in = eval(input("Please enter a positive number"
                            " greater than 1: "))
        if usr_in > 1: break
        else:
            print("Number not valid.")


    prime_list = list_of_primes(usr_in)
    for x in prime_list:
        print(x)


if __name__ == '__main__':
    main()

Answer 1

The problem lies in the if-else within your for-loop:

def is_prime(number):
    if number % 2:
        for num in range(3, int(math.sqrt(number)) + 1, 2):
            print("Testing divisibility by {}".format(num))
            if number % num == 0:
               return False
            else:
                return True
    else:
        return False

Consider the case when number is 25 . In the first iteration of your for-loop, num is 3 . Then, since number%num (ie 25%3 ) is not 0 , False is not returned, and the else is entered and True is returned.

This is what you want to do instead (notice there's no else in the for-loop):

import math


def is_prime(number):
    if number % 2:
        # equivalent to if number % 2 != 0 because if number is
        # divisible by 2 it will return 0, evaluating as 'False'.
        for num in range(3, int(math.sqrt(number)) + 1, 2):
            print("Testing divisibility by {}".format(num))
            if number % num == 0:
                return False
        return True
    else:
        return False

Answer 2

Remove your "is_prime" and "list_of_primes" and replace with these:

def list_of_nonprimes(number):
    return [j for i in range(2, 8) for j in range(i*2, number, i)]

def list_of_primes(number):
    noprimes = list_of_nonprimes(number)
    return [x for x in range(2, number) if x not in noprimes]

... which, by the way, I got from a List Comprehension Tutorial . Also, I don't think you need to import math in Python 3.4+.

Please refer to this post for a very nice explanation of how this works.

Hope this helps!