Unit and Irreducible Numbers

Positive integers: unit, prime, or composite?

There's something very important to note about prime numbers:

$1$ is NOT prime! You might think, wait.... the only divisor of $1$ is $1,$ so it technically meets the requirement for being a prime number, right?

If you choose to interpret it that way, then yes, it does! However, we have chosen to not make $1$ a prime number because of something important we'll learn later, called the Fundamental Theorem of Arithmetic. (If we counted $1$ as a prime number, that Theorem would not be true!)

You could also think of the definition of a prime number as having exactly two factors ($1$ and itself). Then, $1$ isn't prime, since it only has one factor!

Long story short, $1$ is not a prime number, and as we get further into number theory, we'll explain more about why we made this choice. $\square$

Units

Instead of calling $1$ a prime, it is sometimes called a unit. (If you include negatives, then $-1$ is also a unit.) There is an alternative defining property of units worth mentioning: a nonzero integer $u$ is a unit if and only if $u^{-1}$ is itself an integer (rather than fractional). You can see that $u=\pm1$ are the only possibilities; for other nonzero integers $u$, the reciprocal $u^{-1} = \frac{1}{u}$ would be fractional.

Composites

Finally, there are a couple ways to think about composite numbers. On the one hand, a positive integer is composite if it can be factored as the product of two integers $a,b>1$ (note: $a,b>1$ means $a,b$ are not units). On the other hand, a positive integer is composite if it is neither a unit nor a prime. On the third hand, composite numbers are those with three or more divisors.

What is a prime number?

Usually people define prime numbers as "a natural number which is not divisible by any number other than $1$ and the number itself". But this definition has a flaw, it doesn't handle $1$ itself. Here is a better version.

A number $n$ is a prime if and only if it has exactly two positive divisors, not more, not less. Thus, $1$ is not a prime since it has exactly one divisor. $5$ is a prime. But $8$ is not.

To be more precise, we shouldn't actually say that an integer is prime if it has exactly two divisors (if you are familiar with algebraic number theory, you will know what I mean). In fact, in this case, we should say that $n$ is irreducible if $n$ has exactly two divisors. Inspired by Euclid's lemma, we could say that $n$ is a prime if $n$ divides $ab$ implies that $n$ divides $a$ or $n$ divides $b$. This should be the perfect definition of a prime.