On prime(less)ness and composite(less)ness of 1

Question

I was sitting in my room when suddenly my cousin came and asked me, "Why is $1$ neither prime nor composite". Well ofcourse, i was never given an explaination of that in school, it was just a convention. We assumed it. Studying Metric Spaces the same evening I recalled how a set can be open as well as closed. I knew from the beginning of the course

$1)$ Some definition of openness. Later i came accross $2)$ A set is open if its complement is closed.

Can't a similar argument be established for 1 being both prime and composite?

Please give your views on the topic.

Answer 1

The fundamental theorem of arithmetic states that every integer greater than 1 can be represented by a unique (up to reordering) product of prime numbers, and that is an elegant way to state it. But if 1 were prime, then the theorem as stated would be false; $3=3\times 1=3\times 1 \times 1 = \dots$ would be a counterexample. This would make for an uglier fundamental theorem of arithmetic. As far as I know, this is the main reason for the (relatively modern) convention of not including 1 in the set of prime numbers.

EDITED TO ACTUALLY ANSWER YOUR QUESTION

Can't a similar argument be established for 1 being both prime and composite?

No. Because the definition of a prime number is: "a natural number greater than 1 with no positive divisors other than 1 and itself". Furthermore, one way to define a composite number would be: "a composite number is a natural number greater than 1 that is not prime". That is to say, the prime numbers and the composite numbers are mutually exclusive. There are no numbers which are both prime and composite.

Answer 2

I can tell you right off the bat that 1 is most definitely NOT composite.

But, how do you prove that the set of primes and composites is open? The only way I see of doing that is using circular definitions, and I don't know about you, but I consider circular definitions to be a fallacy.

If you want, you can try sending your argument to the author of this page: http://primefan.tripod.com/Prime1ProCon.html though I don't know if he's still updating that page. I think, I'm not sure, that I sent him my argument that prime numbers in $\mathbb{Z}$ have irrational square roots, though he does mention some other properties prime numbers have which 1 lacks. I find these facts far more convincing of 1's non-primality than any hand-wringing about one specific beloved wording of the fundamental theorem of arithmetic.