You've misread the proof. They do not argue that n, n + 1, and n + 2 are co-prime; they only note that n, (n + 1), and n * (n + 1) + 1 are co-prime [and more ge

You've misread the proof. They do not argue that n, n + 1, and n + 2 are co-prime; they only note that n, (n + 1), and n * (n + 1) + 1 are co-prime [and more generally they continue in the same fashion, using the basic fact that n and n + 1 are co-prime for any n]. The proof is sound. However, it is also not particularly different in essence from the classic proof; considering it so is misguided (see https://news.ycombinator.com/item?id=10889290 https://news.ycombinator.com/item?id=10889290).