You&#x27;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].<p>The proof is sound. However, it is also not particularly different in essence from the classic proof; considering it so is misguided (see <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=10889290" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=10889290</a>).