[arachnids]

Here's a slightly more rigorous version of the proof for anyone interested:

1. Suppose the list of primes is finite, and that they are P_1, P_2, P_3 .., P_n

2. Consider the new number P_1 * P_2 * P_3 * .. * P_n + 1. None of the P_is divide this number. Therefore, by definition, it it is a prime.

3. The number we found in (2) is not any of the P_is, and it is a prime. This contradicts the assumption we made in (1). So, the assumption in (1) is wrong, there must be infinitely many primes

The examples in parent's post do not work because they do not follow the framework of this proof. I see parent's point that the claim in the article isn't technically correct, but I think it's reasonable to allow some handwaving in an accessible article written in English :-)