Chapter 9

These are the solutions to the .

1. Here’s one way to use random sampling to determine the ratio of evens to odds:

   ​	​import​ ​random​
   ​
   ​	array = []
   ​	number_of_integers = 1000001
   ​
   ​
   ​	​for​ i ​in​ range(number_of_integers):
   ​	array.append(i)
   ​
   ​	random.shuffle(array)
   ​
   ​	number_of_evens = 0
   ​	number_of_odds = 0
   ​
   ​	random_sample_size = 500
   ​
   ​	​for​ _ ​in​ range(random_sample_size):
   ​	random_index = random.randint(0, number_of_integers - 1)
   ​	​if​ array[random_index] % 2 == 0:
   ​	number_of_evens += 1
   ​	​else​:
   ​	number_of_odds += 1
   ​
   ​	percentage_of_evens = (number_of_evens / random_sample_size) * 100
   ​	percentage_of_odds = (number_of_odds / random_sample_size) * 100
   ​
   ​	​print​([percentage_of_evens, percentage_of_odds])

2. If The Formula—even once—produces a result that isn’t 1, then we know for certain that N is composite.

3. Each time I run The Formula on a prime number, there’s no more than a 50 percent chance that I’ll get a result of 1. Therefore, each time I run the formula it’s akin to flipping a coin. Getting a 1 four times is like a coin landing on heads four times. The odds of this are:

   ​	1/2 * 1/2 * 1/2 * 1/2 = 1/16

   That is, there’s a 1/16 chance that N in this case is prime.