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…
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