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.)
My point exactly: by definition, with no serious reasoning. It's a totally unconvincing word game, and totally lame.
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?
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.
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.
Where did I go astray? Help me out, somebody.