Bonus Question 2

*worth 10 bonus wiki points*

### The Question

Prove the following theorem.

**Theorem.** There are infinitely many prime numbers.

Hint: Use a proof by contradiction. If there were finitely many primes, you could list them as $p_1,p_2,\ldots,p_k$ for some positive integer $k$. To get a contradiction from this, cook up a number $n$ that allows you to *pit the theorem you proved in Problem 3 of Homework 16 against the theorem you proved for Problem 1 of Homework 20*. Just as a reminder:

- Problem 3 of Homework 16 says that every positive integer greater than 1 is divisible by some prime.
- Problem 1 of Homework 20 says that if n and k are positive integers with k > 1, then k cannot divide both n and n+1.

### The Solution

*Proof:*

P1: Assume there are an infinite amount of prime numbers.

### Hints/comments/questions

- Add any questions or comments here, or make a contribution to the solution above…