Why 1 Is the Loneliest (and Least Prime) Number

You ever heard this? Two is the only even prime number. Isn't that wild? Every other prime number than two is odd. Yeah, of course I've heard that. Have you ever heard this one? Five is the only prime number that ends in five. Every other prime number other than five cannot end in a five. Well, yeah, of course, cuz if any other prime number ended in five, it would just be divisible by five. Well, yeah, but that's exactly the point. If any other prime number was a multiple of two, an even number, it would just be divisible by two. We wouldn't call it a prime number at all. This statement has always bothered me. Well, yeah, of course it is. That's what a prime number is. Every single prime number is the only prime number that has that prime number as a factor. By definition, that's what makes a number prime. It can only be factorized as itself times one. What I do like about thinking through prime numbers in this way though is that one thing that often troubles students is how come one isn't a prime number? If we say that prime numbers are numbers that have only themselves and one as a factor, shouldn't we count one as a prime number because it has as itself only one and itself as a factor. It's just that it happens to be the number one itself in the first place. But if you think of prime numbers as the only numbers that have that prime number as a factor, you can see why it would be problematic to let one be a prime number. If one is a prime number, then every single prime number has one as a factor as well. And so it would no longer be true to say that every prime number is the only prime number that has that prime number as a factor. Two is the only prime number that is even. That is that has two as a factor. Three is the only prime number that has three as a factor. Five is the only prime number that has five as its factor and so on. One would not be the only prime number with one as a factor because of course one is actually a factor of every other prime number also. And so that's at least one good thing that comes from this silly statement. It gives us a different way to think through the primes and yet another reason to say confidently one is not a prime