What the Hell? Friday -- On the Primacy of the Number 1

Or not.

In another part of the universe, I was corrected by a math teacher for calling the number "one" a prime.

After a bit of research, I retorted thusly:

I read the info on 1 not a prime and am deeply underwhelmed. Unless I'm missing something big, the reason that 1 is not a prime is that mathematicians have conspired, in a completely arbitrary and discriminatory fashion, to deprive 1 of primehood, by including in the prime definition "greater than 1." If you take a prime to be a number divisible only by 1 and itself, 1 clearly qualifies. Granted it's the degenerate case (chemists will understand what this means) but that really makes no difference.

Tongue in cheek - sure.  But I'm not above a bit of japing.  Anyway, I was advised to have a look at this link.  Which I did.  Here is my new and improved response, re: the four reasons found there (which see.)

Answer 1: 
My point exactly: by definition, with no serious reasoning.  It's a totally unconvincing word game, and totally lame.

Answer 2:
Re: the fundamental theorem - the word "uniquely" is the kicker here.  But that is still just word play.  Tacking on any number of "x1" factors is clearly redundant, and could be eliminated by a more elaborate and linguistically clumsy definition - which I will not attempt at the moment.  BTW, this also illustrates what I meant by a degenerate case.  All the infinite number of potential "x1"s collapse into a single "x1."

Actually, I'll go further and say that the fact that you can write the product non-uniquely is irrelevant to the fact that you can still write it with a single "x1" in the formula, and that THAT is a unique answer.

So we still do not have a really sound reason.

On the other hand, though, lets take another look at the fundamental theorem, "Every positive integer greater than one can be written uniquely as a product of primes."  Ignoring the problematic "uniquely" for the nonce, clearly, if 1 is excluded from the primes, then this theorem crashes and burns.  What are the factors of any prime, say, 5?  They are 5 and . . . and 1!  How can this simple fact not put a Q.E.D. on the primacy of 1?

Answer 3:
By the logic expressed there, I cannot be both a trombonist and a grandfather, though each is a subset of the set  "real humans"  -  being a grandfather is sooo much more important.  Really?  A thing cannot belong to two sets at once?  Five is both a prime and a Fibonacci number.  This reasoning is embarrassingly fatuous.

Answer 4:
This seems to be just an elaborated restatement of definition 3.

Now it's possible that I'm missing something, but here is my verdict.

Reasons for excluding 1 from the primes boil down to, "It's not because I say it's not."   I think that claim has no merit, and can be summarily rejected.

Reason for including 1 in the primes is that without it, the fundamental theorem turns out to be false.   This is not really a big deal for me, but mathematicians will probably find it disconcerting. 

Ergo, 1 is a prime.

Q.E.D.

Where did I go astray?  Help me out, somebody.