Comments by "EebstertheGreat" (@EebstertheGreat) on "Is 1 a Prime Number?" video.

One way to justify the non-primality of 1 is to categorize natural numbers by their order in terms of divisibility. We say a | b ("a divides b") iff there is a natural number n so that an = b. Then we say a < b iff a | b and not b | a. Then you can verify that < is a strict partial order over N, and moreover, that 1 is the least element and 0 is the greatest element. That is to say, for any natural number n, 1 | n and n | 0. For any prime p, we have 1 < p, but there is no other number n for which n < p. So if you draw a graph of <, the 0th level is 1, the 1st level is all the primes, the 2nd level is all the semiprimes, etc., and the ωth level is just 0. Then "composite" just means it's on the (k > 1)th level. In particular, 0 is composite.
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It's not really accurate to say that Gauss was the first to prove the prime number theorem. The unique representation of numbers as products of prime factors was known even to Euclid. To understand why Euclid couldn't present a proof, look at his proof that there are infinitely many primes. It actually just proves that there are at least 4 primes. The point of the proof is that the same reasoning can be applied to any multitude of primes, and thus anyone who understands the proof can see that there are infinitely many primes. Similarly, Euclid's lemma (Elements VII Props. 30-32) can be applied repeatedly to eventually divide any number into prime factors. Euclid IX Prop 14 proves that this decomposition is unique. Many ancient authors clearly assumed this fact, so it was evidently known. Perhaps nobody ever thought to prove it, or had the notation to prove it in full generality, or perhaps those proofs are lost. But Gauss was merely writing down what people like Euler were already using in practice and had been for thousands of years.
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