One is?

One!

Anybody

I mean, yeah. But in a prime/composite definition? It’s neither, the one exception, the golden, whatever.

Why is it not prime? Let’s have a look.

Take 25321, a prime number. 1 is also prime, right? So the factorization of 25321 would be 25321 and 25321*1 at the same time. But we have broken two golden rules:

  1. Every number, even prime numbers, have exactly one prime factorization
  2. A number cannot be both prime and composite at the same time.

Also, a proof that came in my head while writing:

You know Euclid’s Theorem that states ‘there is no prime because of this, that and stuff’? Full Proof:

Say the list of primes were finite, like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and that’s it. Multiply them all together, add one, it’s 6469693231 (2*3*5*7*11*13*17*19*23*29+1). Since this number is bigger than the “last” prime, it must be composite (some cases are when it actually is prime). But if you divide it by any “Prime” number, it will result in a remainder 1. So there must be some other “prime” out there that is prime.

Curtains, lights, action!

Say that one was prime, then Euclid’s Theorem to primes to 30 multiplication would still be 6469693231. But since 1 can divide 6469693231 evenly, and one is “prime”, Euclid’s Theorem has this weird exception, and that contradicts the fact that there are infinitely many primes.

1 isn’t composite either, since it doesn’t have any prime numbers that can divide it evenly.

So there you have it.

1 is not prime. So it’s not composite! Yay!

Q.E.D.

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