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Comment by phasmantistes | Hacker News Reader

That's not really an accurate representation of the old proof. You've turned it around to make it look like the new one, but that's not how it is originally laid out at all.

The fundamental problem (for constructivists) with Euclid's proof is that it assumes the existence of a set of all primes, and then proves that that set is incorrect (either contains a composite, or doesn't contain at least one specific prime).

Saidak's proof, on the other hand, only assumes the existence of a single integer, and then constructs infinitely many primes from that integer.

Your restatement of Euclid's proof is basically "look, Euclid and Saidak proved the same thing", not "look, Euclid's proof is constructive too".

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Two points:

A) Euclid never assumes the existence of a finite set of all primes (doing so at the outset is a framing adopted only by later mathematicians); people just frequently erroneously think Euclid's proof did this, never having actually read it.

Rather, Euclid constructively frames things, directly arguing that, for any finite set of primes, there is furthermore some prime outside that set. See http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht....

B) Regardless, I don't particularly care how Euclid phrased his proof; I never mentioned Euclid. Certainly, mathematicians have been aware that Euclid's insight could be framed in the way I called an "old proof" for a long, long time. The distinctions between the various framings here are trivial.

I think the point is you can easily remove the "deficiencies" of Euclid's proof. Basically, say: let p_1, ..., p_n be the first n primes. We can construct a new prime by examining p_1...p_n + 1. This process can be continued indefinitely.

In fact, I think the statement "this process can be continued indefinitely" is quite misleading -- to formalize this, I think you need proof by contradiction. So, the difference is the syntax used, not the actual semantics of the proof.

There a ways to formulate "the infinity of the prime numbers" without any recourse to a previously-constructed set of all prime numbers (or even all natural numbers). Two such ways are:

1. "For any natural number n, there is a prime number greater than n."

2. "For any natural number m, and for any list of m prime numbers p_1, ..., p_m, there exists a prime number not on that list."

Both statements have a perfectly fine constructive proof. The second statement is more or less precisely what Euclid proved. The first statement follows from the second one with a bit more work.

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