Old proof: Let&#x27;s produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (some prime factor of (v_n + 1)).<p>&quot;New&quot; proof: Let&#x27;s produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (v_n + 1).<p>Doesn&#x27;t seem terribly different or more constructive or any such thing to me. I&#x27;d say both of these have the same fundamental content, just framed slightly differently.<p>[Just to head off what might superficially seem to be a significant distinction: Note that, in both proofs, we use the (constructive) fact that every integer &gt; 1 has a prime factor, and even in the &quot;new&quot; proof, in order to actually extract an infinite stream of primes, one must actually carry out this process of finding prime factors for integers &gt; 1 on demand.]