At Saturday Course we were working with prime numbers, and one fifth-grader asked his classmates a question:
Student A: Is 1 a prime number?
Student B: No.
Student A: So it’s composite?
Student B: No, it isn’t prime and it isn’t composite.
Student A: Why isn’t it prime?
Student B: Because it only has one factor. Prime numbers have two factors.
Me: That’s true, but it’s not the real reason.
Clearly Student B’s teacher had given him this strange definition. I call it “strange” because it’s unmotivated: why would anyone want to define “prime” in that way? What’s so special about having two factors (or, preferably, divisors)? Why would two factors make a number prime, whereas one or three does not? What sense does that make?
The real reason that 1 can’t be prime is that the Fundamental Theorem of Arithmetic (FTA) wouldn’t hold if it were. Number theory guarantees that every positive integer has a unique prime factorization (ignoring order). For example, 40 equals 23×5 and nothing else. But if 1 were prime, 40 could also be 23×5×1 or 23×5×12 or 23×5×13, etc. That’s why we exclude 1 from being prime. And there’s an important lesson here: mathematicians often tweak their definitions in order to make theorems work. In the words of…well, I can’t remember who said this…they shoot the arrow and then draw the target.