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16 def primeGen(n): 17 for i in xrange(2, n): 18 prime = True 19 if i % 2 == 0 and i != 2: 20 continue 21 sqrtp = int(i ** 1 / 2) 22 for j in xrange(2, sqrtp): 23 if j % 2 == 0: 24 continue 25 if i % j == 0: 26 prime = False 27 break 28 if prime: 29 yield i

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20 def prime(num): 21 # num is actually a string because input() returns strings. We'll convert it to int 22 num = int(num) 23 24 if num < 0: 25 print("Negative integers can not be prime") 26 quit() 27 if num is 1: 28 print("1 is neither prime nor composite") 29 # See how I lazily terminated program otherwise it'd forward "None"(default behaviour of python when function 30 # returns nothing) rather than True or False. Which could mess up the program. 31 # If we hit this if statement above statement is printed then program exits. 32 quit() # Now you don't need to get sys.exit() to exit python has quit to handle the same thing 33 if num in [2, 3]: 34 # if given argument is 2 or 3, it is prime. We used list without defining a variable which is perfectly valid 35 return True 36 if num % 2 == 0: # excluding all even numbers except two. 37 return False 38 else: 39 # Here we are starting counter variable from 3 in range. Second argument excludes numbers above one third 40 # of the given argument. Third argument in range sets steps to take to 2. This makes loop to iterate odds 41 for x in range(3, int(num/3), 2): 42 # Checking if argument is divisible by counter. % is modulus operator which returns remainder of division 43 if num % x == 0: 44 return False 45 # It's okay to have more than one return statement when program hits return statement it exits the function. 46 return True

12 def primes(): 13 yield 2 14 yield 3 15 16 for i in itertools.count(start=5, step=2): 17 if is_prime(i): 18 yield i

12 def genprime(n): 13 global isprime 14 isprime = [True] * (n+1) 15 sN = int(math.floor(math.sqrt(n))) 16 17 for i in range(3, n+1, 2): 18 if (isprime[i]): 19 yield i 20 if (i < sN): 21 ni = 2*i 22 while (ni <= n): 23 isprime[ni] = False 24 ni += i

7 def isPrime(num): 8 # Returns True if num is a prime number, otherwise False. 9 10 # Note: Generally, isPrime() is slower than primeSieve(). 11 12 # all numbers less than 2 are not prime 13 if num < 2: 14 return False 15 16 # see if num is divisible by any number up to the square root of num 17 for i in range(2, int(math.sqrt(num)) + 1): 18 if num % i == 0: 19 return False 20 return True

59 def gen_prime(nbits, rand): 60 while True: 61 p = rand.get_int_bits(nbits) 62 p |= 2**nbits | 1 63 if miller_rabin(p, rand.randint): 64 return p

50 def prime_factors(n): 51 """Lists prime factors of a given natural integer, from greatest to smallest 52 :param n: Natural integer 53 :rtype : list of all prime factors of the given natural n 54 """ 55 i = 2 56 while i <= sqrt(n): 57 if n % i == 0: 58 l = prime_factors(n/i) 59 l.append(i) 60 return l 61 i += 1 62 return [n] # n is prime

6 def is_prime(n): 7 ''' 8 checks if a number is prime 9 ''' 10 if n < 2: 11 return False 12 if n == 2: 13 return True 14 for x in range(2, int(n**0.5)+1, 2): 15 if n % x == 0: 16 return False 17 return True

21 def is_prime(n): 22 for i in xrange(2, int(sqrt(n)) + 1): 23 if n % i == 0: 24 return False 25 26 return True

3 def is_prime(n): 4 if n <= 1: 5 return False 6 elif n == 2: 7 return True 8 elif n % 2 == 0: 9 return False 10 for i in xrange(3, int(sqrt(n))+1, 2): 11 if n % i == 0: 12 return False 13 return True