What if we permitted 1 to be prime? In that case, 84 would also have the "prime" factorisation 1 x 1 x 1 x 2 x 2 x 3 x 7. That is, 84 could still be factorised, but it would no longer have a unique prime factorisation.
I did read the article. I was just pointing out that the poster's original statement seemed to exclude prime integers -- well, unless the empty product is a product of primes.
What if we permitted 1 to be prime? In that case, 84 would also have the "prime" factorisation 1 x 1 x 1 x 2 x 2 x 3 x 7. That is, 84 could still be factorised, but it would no longer have a unique prime factorisation.