Fiddling with Euclid's Proof

When you add relatively prime numbers you get an answer which is relatively prime to each of the starting numbers.  This means in Lego you get all new colours. We can use this to prove that there are infinitely many primes.  It is very similar to Euclid's proof, but it 'generates' more primes.

You start by assuming there are only finitely many primes - in Lego take one block of each colour. Then make two towers and add them together. The answer must always be made out of new colours.

Here we start with the blocks for 2, 3, & 5, and end up with new blocks for 17, 13, & 11.