Since time immemorial, humans have asked: Why are we here? What is our purpose?
I assure you that I have a beautiful and exhaustive answer to that question, which this blog post is unfortunately too small to contain, so let us instead discuss a question almost as rich and immortal: Why doesn’t 1 count as a prime number?
(The key to success, my friends: attainable goals.)
First, the typical student’s answer:
The student points to the common definition, something along the lines of: “a prime number is divisible by precisely two numbers: 1 and itself.” Since 1 is not divisible by two numbers, it is not prime.
This answer amounts to “because I said so.” Or perhaps more sympathetically, “Because my teacher and/or textbook said so.” It is an answer all about deference to power. It is not a wrong explanation, so much as a failure of explanation. Yes, the definition says so, but why does it say so?
Ask a more conspiracy–minded person, and you’ll hear something like:
To be honest, I get where the “1 is prime” truthers are coming from. If a prime is “divisible only by 1 and itself,” then hey, doesn’t the number 1 count? Why the arbitrary “it needs at least two factors” part of the rule?
These people are more fun to talk to. They have come to their own conclusions, via their own critical thought. As a bonus, they often have strong opinions about the whereabouts of Tupac and Bigfoot.
After that conversation, you’ll be thoroughly confused, so it’s time to check with a linguist. Why isn’t 1 prime?
It’s a little weird, isn’t it, that “prime” means “first,” and yet the first number isn’t prime?
(It’s almost as weird as the fact that, in the U.S., we call the first derivative “prime,” and the second derivative “double prime” – literally, “double first.”)
Well, here’s a fact I learned recently from Tom Edgar: In the 19th century, many mathematicians did consider 1 to be prime! The irreplaceable Evelyn Lamb explains the details, including the juicy fact that no less an authority than G.H. Hardy listed 1 among the primes.
It’s more fodder for the conspiracy theorists, and a troubling fact for the student. If definitions are sacrosanct, then why are they subject to the whims of history?
Well, let’s hear why 1 isn’t prime from the real experts: the band Three Dog Night.
(Props also to Harry Nilsson, who wrote the song. And now that it’s in your head, I’ll save you the search: here’s the YouTube link.)
Okay, fine, here’s the reason 1 isn’t prime, from the mathematician:
For a mathematician, the definition of prime numbers doesn’t stand alone, like some kind of definition passed down from on high. It’s part of our theory of factorization. As Chris Caldwell and Yen Xiong put it (hat tip to Evelyn Lamb for the quote):
Whether or not a number (especially unity [i.e. 1]) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
A beautiful feature of the whole numbers is that each can be factorized in precisely one way. I mean, yes, sure, a number like 30 can be factorized in several ways:
- 6 * 5
- 10 * 3
- 15 * 2
But in each case, the first factor can be broken down further, yielding the same ultimate factorization:
- 2 * 3 * 5
- 2 * 5 * 3
- 3 * 5 * 2
2, 3, and 5 are the prime factors of 30. Each can be broken down no further; they are 30’s fundamental constituent elements, its DNA. But if you allow 1’s… well, then you could extend the factorizations arbitrarily:
- 2 * 3 * 5 * 1
- 2 * 3 * 5 * 1 * 1
- 2 * 3 * 5 * 1 * 1 * 1
The 1’s add no information. And so it’s more convenient to exclude 1 from the list of prime numbers. That’s what allows us to say that each number has a unique prime factorization.
In the end, the mathematician and the student both point to the definition. The difference is that the novice treats definitions as fixed, eternal, and immutable; the mathematician does not.
Math does indeed present us with a fixed, eternal, immutable world. But the definitions aren’t part of it. They’re the signposts we’ve added for navigating this world, the trail-heads we’ve identified. Some make for easier paths than others. But there are always new trails to blaze.