How to generate prime palindromes in Python 3

Question

I'm trying to generate the prime palindromes between two integers (given by the user) and simply can't figure out where I'm going wrong. Please can someone explain?

Here's what I've got so far:

#Palindrome test
def palindrome(num):
    str(num) == str(num)[::-1]

#Prime test (excl. 3, 5 & 7 because not palindrome)
def prime(num):
    abs(num)%2 != 0
    abs(num)%3 != 0
    abs(num)%5 != 0
    abs(num)%7 != 0


N = int(input("Enter the start point N: "))
M = int(input("Enter the end point M: "))

for x in range(N, M):
    if palindrome(x) and prime(x):
        print(x)

Alright, as was pointed out - there's more than one issue in the code above. I've tried again and come up with this (below), which manages to print out all the palindromes, but also includes the non-prime numbers. Where am I going wrong?

N = int(input("Enter the start point N: "))
M = int(input("Enter the end point M: "))

def palindrome(num):
    return str(num) == str(num)[::-1]

def prime(x):
    for x in range(N, M+1):
        for y in range(2, x):
            if x%y != 0:
                return True

for z in range(N, M):
    if palindrome(z) and prime(z):
        print(z)

Thanks for all the feedback so far!

Answer 1

I took @Banach code little further. Test for prime is changed to accommodate numbers that are divisible by prime numbers itself. Inspiration from here.

The prime test now works for much larger numbers as well.

Demo Code

#Palindrome test
def palindrome(num):
    return str(num) == str(num)[::-1]

#Prime Test
def is_prime(n):
  if n == 2 or n == 3: return True
  if n < 2 or n%2 == 0: return False
  if n < 9: return True
  if n%3 == 0: return False
  r = int(n**0.5)
  f = 5
  while f <= r:    
    if n%f == 0: return False
    if n%(f+2) == 0: return False
    f +=6
  return True

#Get user input
N = int(input("Enter the start point N: "))
M = int(input("Enter the end point M: "))

for x in range(N, M):

    #Check for palindrome and prime number
    if palindrome(x) and is_prime(x):

        #Omitting 2,3,5,7 as per request
        if abs(x)%2 != 0 and abs(x)%3 != 0 and abs(x)%5 != 0 and  abs(x)%7 != 0:
            print(x)

Output

Python 2.7.9 (default, Dec 10 2014, 12:24:55) [MSC v.1500 32 bit (Intel)] on win32
Type "copyright", "credits" or "license()" for more information.
>>> ================================ RESTART ================================
>>> 
Enter the start point N: 2
Enter the end point M: 20000
11
101
131
151
181
191
313
353
373
383
727
757
787
797
919
929
10301
10501
10601
11311
11411
12421
12721
12821
13331
13831
13931
14341
14741
15451
15551
16061
16361
16561
16661
17471
17971
18181
18481
19391
19891
19991
>>>