python中的素数代码是如何工作的？

Question

def is_prime(num):
    for i in range(2,num):
        if (num % i) == 0:
            return False
    return True

def all_primes(num):
    primes = []
    for n in range(2,num+1):
        if is_prime(n) is True:
            primes.append(n)
    return primes

num = int(input("Enter upper limit: "))
primes = all_primes(num)
print(primes)

How does it happens... What understand is if I enter 10 It will be 10%2= 5...the remainder is 0 so it skip to the next one 9%2 = True there is remainder.

It moves to the next one 9%3 remainder is 0 so it skips to 8% ...,then 7 .......but what I dont understand is 7 if it's check until 7%7 ==0 then how did it add 7 as Prime number???

I'm so confused.

Thank you for any comment

Answer 1

but what I dont understand is 7 if it's check until 7%7 ==0

The range function from the following line:

for i in range(2, num):

goes from 2 to num - 1 .

So, the range() function doesn't include the last (stop) number in the result.

Answer 2

You got it wrong. If you enter 10 , it wouln't check 10%2 and then 9%2 ...

It would check 10%2 , then 10%3 , then 10%4 ... ie it would check 10%i , for i ranging from 2 to 9 . However, note that execution in is_prime stops as soon as the remainder is 0:

for i in range(2,num):
        if (num % i) == 0:
            return False

So if num is 10 , only 10%2 would be checked, and since it's 0 , it would immediately return false .

If num is 9 , for example, it would check 9%2 . Since the remainder is not 0 , it would move up to checking 9%3 . The remainder is 0 there, so it would return false in that case.

If num was 7 , it would check 7%2 . The remainder is not 0 so it checks 7%3 , which is also different from 0 , and so on. The last check performed would be 7%6 , because for i in range(2,num) means that it will iterate with i ranging from 2 to num-1 . In this case, is_prime(7) would return true because 7%i , with i ranging from 2 to 6 is different than 0 .

Answer 3

A prime number, by definition, is a number that is divisible only by itself and 1.

Therefore, your function is_prime gets the number you want to check if it is prime, and try to find any other number that can divide it inside the interval [2, num), this interval is implemented using the function range(2,num) inside the for loop.

If it finds any case, ie, if the if (num % i) == 0 is satisfied, we know that there is a number which divides it, therefore, by definition it is not a prime number. After checking the whole list, if you cannot find a divisor, we can say that the number is prime, ie, the return True on your function.

Example:
You want to check 9
(1st step) num = 9, i = 2. 9%2 != 0, we continue in the loop with next value of i.
(2nd step) num = 9, i = 3. 9%3 == 0, we return False. As we know, 9 can divide by 3.

Another example:
You want to check 7
num = 7, i = 2 7%2 != 0, so we continue in the loop with the next value of i
num = 7, i = 3 7%3 != 0, so we continue in the loop with the next value of i
num = 7, i = 4 7%4 != 0, so we continue in the loop with the next value of i
num = 7, i = 5 7%5 != 0, so we continue in the loop with the next value of i
num = 7, i = 6 7%6 != 0, so we continue in the loop with the next value of i

The if was never fulfilled, therefore we reach the end of the loop and 
call the next command, which is return True. 

Answer 4

when is_prime is called for number 7 , the loop will run from 2 to 6 only. Hence 7%7 will never be checked , returning True for such case.

Answer 5

Let's consider the two function separately:

def is_prime(num):
    for i in range(2,num):
        if (num % i) == 0:
            return False
    return True

This works like this:

• let's consider a number num
• get a variable i to assume the value 2
• evaluate the condition num % i == 0
• if the condition is met, then exit from the function and return False
• otherwise increment i by 1 and repeat the last two steps until i is equal to num - 1
• exit the function and return True

So, this never gets to evaluate 7 % 7 : when num is 7 the last value of i is 6 .

---

def all_primes(num):
    primes = []
    for n in range(2,num+1):
        if is_prime(n) is True:
            primes.append(n)
    return primes

This gets a number num which is the upper limit of the numbers to consider, then:

• it creates an empty list named primes
• it loops n from 2 until num + 1 excluded (or num included)
• if the result of is_prime(n) is True, it appends the number to the list .
• finally, return the list in primes .

---

Finally the main body of the script:

num = int(input("Enter upper limit: "))
primes = all_primes(num)
print(primes)

gets you a string from the keyboard, convert it to int and assign to num , then call all_primes(num) and assign its result to primes and print the list to screen.

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So, when you input 10 from the keyboard, it will check the values from 2 to 10 for being prime with is_prime() , and not from 10 and reducing them, as you seems to suggest.