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.
The UFD argument is probably the best for everyone, since it only involves FTA.
Depending on who’s asking, one nice thing to consider is other rings (of numbers) where the role of units become a bit more involved e.g. Gaussian integers or adjoining a root (such that you obtain a UFD). It becomes clearer with examples that the dichotomy of prime and composite should in fact be a trichotomy of prime, composite and unit.
I never found the argument that because 1 does not generate a prime ideal to be unpersuasive, since we exclude the entire ring from being a prime ideal by *definition*.
Since you omitted the word “positive” from the various definitions, can I play devil’s advocate and claim that 1 must be prime because it can be divided by precisely two numbers: 1 and -1. 🙂
Reblogged this on Trasnochada and commented:
Un magnífico post sobre la condición de primo (o no), del número 1
Put the prime in the coconut and drink it all up.
I’m comfortable saying that we need 1 not prime for all the lovely math that follows. 1 is special enough, it’s the next addend and the multiplicative identity.
I am a part of the ultra conspiracy camp. 1 is prime, but it is not a number.
“That’s what allows us to say that each number has a unique prime factorization.” By that definition 1 is not a number since 1 does not have a unique prime factorization; the same is true for 0 as well.
1 does have a unique prime factorisation: the empty list is a list of primes and its product is indeed the identity of multiplication, 1 (just as the sum of the empty list is the additive identity 0; in each case, associativity says that the sum or product of the concatenation of two lists is the sum or product of the respective lists’ sums or products, so the sum of the empty list must be the identity, in each case, since its concatenation with any list is that list). For any positive counting number, there is a unique monotonically non-decreasing list of primes whose product is that number; this is its prime factorisation; the multiplicity of each prime as a factor of the number is the number of times the prime shows up in this list. In the case of the empty list, as factorisation of 1, every prime’s multiplicity is zero.
However, indeed, 0 does not have a prime factorisation, much less a unique one. It has every prime as a factor, as many times as you like. Indeed, if you consider the least common multiple – normally at least as big as each factor – of any number and zero, the answer is zero; while the highest common factor – usually no greater than any of its multiples – of any number and zero is the given number. In this sense, zero is greater than all other whole numbers; for multiplicative purposes, zero is infinity, the product of arbitrarily many of each and every prime all at once.
Awesome post
I’m not a mathematician!!! Why the heck should I care???
Sorry I’m late–have I been doing this wrong? I thought prime numbers were only positive, and that the definition of prime numbers is “any number greater than one divisible only by itself and one.”
Mathematics is a game, and we choose the rules/definitions to make the game, fun and productive. It is a better game when 1 is not a prime, because the unique factorization theorem is so much fun. Some changes to rules/definitions would allow a similarly good outcome, and thus could be considered (and some people can argue that making 1 prime is one of these) and some changes are just really bad. By analogy, we could change the rules of Monopoly so that you collect $220 dollars for passing Go, and that would be an equally good game, but changing the rules so that you pay $500 dollars or get $1,000,000 for passing Go would make the game just plain bad.
This is a case of what my advisor Silvio Micali often says: “the right definition is the one that lets you prove the theorem you want.”
This is a really nice explanation. Thanks!
Late to the game, but I would like to play Devil’s Advocate for a second.
The argument that 1 is not prime because it maintains the uniqueness of prime factorisation’s (UPF) seems to be the most popular in the comments, which I fully understand.
But plenty of things in maths are defined/proved as unique up to isomorphism (such as equivalence classes or orbits in a group). So why would the UPF be any less useful/powerful if we declared them to be unique up to identity?
A prime number has ONLY two factors, 1 and “another number” and the product is the “another number”.