# Determine if number is a prime number

I made a function that takes an argument x and returns True if x is a prime number and False if x is a non-prime number.

I am a beginner and think the code can be improved. I had to add the else statement because for is_prime(2), I kept getting None instead of True. I think this was because when x is 2, list_n = range(2,2) prints [].

Are there ways I could change this so there is not a separate else statement?

def is_prime(x):
list_n = range(2,x)

if x <= 1:
return False
elif x-1>2:
for n in range(len(list_n)):
list_n[n] = x % list_n[n]
for n in list_n:
if n == 0:
return False
for n in list_n:
if n > 0:
return True
else:
return True


To begin with, you don't need list_n at all: a decision that the number is composite can be made on the fly, as soon as some remainder becomes 0. If none of them were 0, you know that the number is prime:

    for n in range(2, x):
if x % n == 0:
return False
return True


is effectively what your code is doing, but without the list, and without special case for 2.

That said, of course the algorithm could be greatly improved, but it is beyond the point of this exercise.

• Don't forget the edge case n <= 1, which was handled in original code. Jun 18, 2017 at 15:41
• n.b. that for n in range(2, x) checks more numbers than necessary to see if x is prime. The greatest unique divisor of x (that isn't a co-divisor of a smaller number, e.g. 32 => (2, 16)) is sqrt(x). for n in range(2, int(x ** 0.5)+1) is sufficient to test for primality. Jun 18, 2017 at 17:08
• @AdamSmith Yes, this is what the last line in my post is about.
– vnp
Jun 18, 2017 at 17:10
• @vnp Ah. I assumed you were talking about using a Sieve or etc Jun 18, 2017 at 17:13
1. You don't need to create a list. Just iterate over potential divisors:
for d in xrange(2, x): ...

2. The condition x - 1 > 2 seems kind of complicated. I suggest adding a guard at the beginning of the function:

if n <= 1:
return False


and using a generic procedure for all other numbers (including 2. It works for x = 2. There's nothing special about it). It'll make your code easier to follow.

3. You can make it more efficient by checking the divisors only up to sqrt(x) inclusively. If a number is not a prime, it must have at least one divisor not greater then its root.

• Your first point depends on whether he's using Python 2 or Python 3. Python 3 has no xrange, since it's range already behaves as such.
– Mast
Jun 18, 2017 at 17:50
• @Mast He must be using python 2. Otherwise, list_n[n] = ... won't work (range doesn't support assignment in python 3) and his code is broken for any x >= 3. Jun 18, 2017 at 17:58
• I'm so accustomed to Python 2 I didn't even catch that would break on 3. Good call.
– Mast
Jun 18, 2017 at 18:00
1. Use a better variable name than x.
2. Add separate if-clauses to check if your input number is <= 1, == 2 and % 2 == 0.
3. Beyond the above exclusions, you only have to deal with odd numbers. Check only against odd divisors: xrange(3, sqrt(your_input), 2).
4. This loop is entirely useless:

for n in list_n:
if n > 0:
return True


You'll end up with something like

def is_prime(integer):
"""Returns True if input is a prime number"""
if integer < 2 or not isinstance(integer, (int, long)):  # remove long if using Python 3
return False
if integer == 2:
return True
if integer % 2 == 0: # which can also be written as -> if not integer & 1:
return False
for divisor in xrange(3, int(math.sqrt(integer)) + 1, 2):
if integer % divisor == 0:
return False
return True

• I must say I disagree with this. Calling a variable x where it is clear that this is an abstract variable (yes, an integer) is much more useful than calling it integer. If you see a function called add(a, b) isn't it obvious what a and b do? Jun 17, 2017 at 21:34
• @Coal_ yes, but nearly all pep8 based linters will report that as an invalid argument name. Jun 18, 2017 at 5:23
• @hjpotter92 then fix the linter. single-letter variable names are made for instances like this, and PEP8 does not declare against them. Sounds like an overreach of the linter. Jun 18, 2017 at 17:14

The third for loop can be replaced with a single return True statement (because it's the exact opposite of the second for). Also, the else: can be removed as if any previous conditions were satisfied, the function would have already returned. The return True block formerly in the else: block can be fused with the third for loop's return, producing this code:

def is_prime(x):
list_n = range(2,x)

if x <= 1:
return False
elif x-1 > 2:
for n in range(len(list_n)):
list_n[n] = x % list_n[n]
for n in list_n:
if n == 0:
return False
return True


Another suggestion is to replace the first for with a list comprehension. This is left as an exercise to the reader.

Just remove else: and unindent return True. That should work since if and elif will always return if entered.