Is 1 Prime, and Does It Matter?

If you ask a person on the street whether 1 is a prime number, they’ll probably pause, try to remember what they were taught, and say “no” (or “yes” or “I don’t remember”). Or maybe they’ll cross the street in a hurry. On the other hand, if you ask a mathematician, there’s a good chance they’ll say “That’s an excellent question” or “It’s kind of an interesting story…”

Some people treat the non-primeness of 1 as a mathematical fact and nothing more, but those people are missing out on something important about the nature of mathematics.

THE PREHISTORY OF PRIMES

In early days, 1 wasn’t universally regarded as a number at all. For the Pythagoreans, the first counting number was 2; 1 was the Unit from which all the numbers (2, 3, etc.) were built. So 1, not being a number, was certainly not a prime number. Euclid, although not a member of the Pythagorean Order, agreed that the first prime number was 2.

But Greek thought wasn’t homogeneous. Some, such as Plato’s nephew Speussippus, thought that 1 was not only a number, but a prime number at that. So controversy about the status of 1 has a respectable pedigree.

Nor can the practice of calling 1 a prime be complacently relegated to the midden of ancient, long-discarded mistakes. The great Leonhard Euler, the pre-eminent mathematician of the eighteenth century, treated 1 as a prime in his correspondence with number-theorist Christian Goldbach. Even in the twentieth century, the mathematician G. H. Hardy, coauthor of the first great work on number theory written in the English language, classified 1 as a prime in his early writings.

Were Euler and Hardy being stupid or careless? Far from it. They were doing what good mathematicians always do: maintaining a flexible attitude toward terminology, and keeping in mind that sometimes the right way to define things only comes into focus when you’ve played with several variants.

So if your attitude toward my title was “Yeah, why does it matter?” you’re asking a question that Euler and Hardy – who both sometimes included 1 among the primes and sometimes didn’t – would have endorsed. After all, the number 1 has many properties in common with the primes.¹

But you shouldn’t get the idea that in the modern era there’s disagreement about the status of 1; by universal consensus, 1 isn’t a prime.² Does that mean we’re forced to classify 1 as a composite number, i.e., a factorable number like 4, 6, 8, and 9? Or is there a third possibility?

THE LONELIEST NUMBER

In the preface to his 1914 table of primes, the number theorist D. N. Lehmer, by way of justifying his decision to include 1 in the table, pointed out that “the number 1 is certainly not composite in the same sense as the number 6,” adding “if it is ruled out of the list of primes it is necessary to create a particular class for this number alone.” For Lehmer, that was sufficient reason to list 1 as a prime; leaving 1 out in the cold, calling it neither prime nor composite, didn’t seem like an option.

1 is certainly an exceptional number for many reasons. One distinctive property of the number 1 is that it’s its own reciprocal. No other positive integer has this property. When we enlarge our number system to include zero and the negative integers, 1 acquires a buddy in the person of its negative, the number −1, which, like 1, is its own reciprocal. Further enlarging our scope to include the rational numbers and the real numbers brings us no new numbers with this property. But when we enlarge yet again, to the complex numbers, although we don’t get any new numbers that are their own reciprocals, we get two numbers that are simultaneously each other’s negatives and each other’s reciprocals: i and −i.

Just as the integers form an interesting subsystem of the real numbers, the Gaussian integers — complex numbers of the form a + bi where a and b are ordinary integers — form an interesting subsystem of the complex numbers. The Gaussian integers taken in aggregate form what mathematicians call an integral domain (in this essay I’ll use the shorter term “domain” for brevity) in which numbers can safely be added, subtracted, or multiplied without ever leaving the domain. Notice that I left division off the list of safe operations; in a domain, you usually can’t divide one element by another. But when a special element of a domain — call it u — has the property that the reciprocal of u also belongs to that domain, then every element of the domain can be divided by u: just multiply that element by the reciprocal of u. In the domain of integers, the only such elements are u = 1 and u = −1, but in the domain of Gaussian integers, there are four of them: 1, −1, i and −i.

An even more interesting example is the domain consisting of all numbers of the form a + b sqrt(2) where again a and b are ordinary integers. In this domain there are infinitely many numbers whose reciprocals belong to that same domain: for instance, 1 + sqrt(2) and −1 + sqrt(2) are each other’s reciprocals, 3 + 2 sqrt(2) and 3 − 2 sqrt(2) are each other’s reciprocals, and so on.

The study of such number systems, pioneered by Carl-Friedrich Gauss and now a thriving specialty in its own right, is called algebraic number theory. In this subject, numbers in the domain whose reciprocals also belong to the domain are called units. 1 is no longer lonely; it has a hip-and-happening club to belong to.³

So those Pythagoreans from the start of this essay were onto something. From a modern perspective, they were right in singling out 1 for special treatment and insisting that we pay deference to 1 as a Unit; but whereas they viewed being-a-Unit as incompatible with being-a-number, we regard 1 as both a unit and a number.

AVOIDING AWKWARDNESS

I suspect one reason Lehmer persisted in calling 1 prime is etymological. The Greeks called the primes the protoi arithmoi or “first numbers”, and the Latin word “primus”, from which we derive the words “prime” and “primary”, has similar connotations. How can 1 be the first number we say when we count, and yet not be counted as one of the First Numbers?

But even before Lehmer classified 1 as a prime, most modern mathematicians had quietly decided it didn’t deserve that designation. This consensus arose not because of any one thing, but because of dozens of different ways in which treating 1 as a prime led to awkwardness.

A case in point is the sieve of Eratosthenes, mentioned by Lehmer on the same page as his argument for calling 1 a prime. Lehmer writes: “Eratosthenes, a contemporary of Euclid, was the inventor of a ‘sieve’ process for removing the composite numbers from the series of natural numbers. He first wrote the numbers in order, and then removed the multiples of 2 by erasing every other number after 2. He then erased every third number after 3, then every fifth after 5, and so on. In this way, by rejecting the multiples of the successive unerased numbers, he obtained the series of primes.”

Let’s break this down. First I’ll erase, or rather shade out, the multiples of 2 (not including 2 itself, of course) between 1 and 25:

Then I’ll get rid of the multiples of 3:

Then the multiples of 5:

And so on.⁴

But hang on a minute. If we’re supposed to remove the multiples of each successive number that hasn’t been removed yet, shouldn’t we start the game by removing all the multiples of 1? Of course, then the game would end very quickly, and 1 would be declared the only prime.

Of course that’s not how the sieve works. We treat 1 in a different way than 2, 3, 5, etc.; specifically, we don’t cross out all its multiples. If we insist on calling it a prime anyway, we must admit it’s a very special prime.

Another case in point is the Fundamental Theorem of Arithmetic, otherwise known as the uniqueness-of-factorization theorem.⁵ Every composite number can be written as a product of primes, and moreover, there’s only one way to do it, if we agree to ignore the order in which the factors appear, so that for instance 2 × 3 and 3 × 2 count as the same factorization of 6.⁶ If 1 were classified as a prime, then there’d be more than one way — infinitely many ways, in fact — to write 6 as a product of primes: 2×3 and 1×2×3 and 1×1×2×3 and so on.

Of course a fervent 1-is-prime holdout could stand his ground and rephrase the Fundamental Theorem of Arithmetic to allow for this, so that two factorizations that differ only in the ordering of the factors, or in the inclusion of a different number of factors equal to 1, would still count as the same. Then the deviant definition of primeness that includes 1 as a prime would still permit him to formulate the uniqueness of factorization theorem, but at the cost of some awkwardness.

PUTTING THEOREMS FIRST

I don’t know of any modern-day 1-is-prime holdouts, but I imagine that Christian Goldbach, the correspondent of Euler whom I mentioned before, would have held onto the idea longer than most of his contemporaries. Goldbach is mostly known nowadays for coming up with the celebrated and still-unproved conjecture that every even number bigger than 2 can be written as a sum of two primes. Or at least, that’s how we phrase it nowadays. Goldbach himself conjectured that every even number (meaning, every even positive integer) can be written as a sum of two primes, including the even number 2, because 2 can be written as 1+1, and for him, 1 was prime.

If I were at a party with Goldbach and we were debating the proper definition of “prime“, I’d be forced to admit that his conjecture is more easily stated using his more inclusive definition of the word, but I’d tell him that his conjecture is one of the few cases in which treating 1 as a prime makes things simpler; more often, it makes things more complicated. “You couldn’t have known this,” I tell him, “because the theorems that make it more natural not to call 1 prime still lay in the future when you did your work.”

“What theorems?” asks Goldbach, and I proceed to tell him about a few, taking special pleasure in introducing him to Gauss’ Law of Quadratic Reciprocity. Quadratic reciprocity is a beautiful fact that relates mod p arithmetic with mod q arithmetic whenever p and q are two different odd primes, that is, two different primes bigger than 2.⁷ In telling Goldbach the story behind this theorem, I’m careful to use the phrase “odd prime bigger than 1” to talk about the things that I would call (more simply) “odd primes”, so as not to confuse him.

At this point, Nicomachus of Gerasa, who is at the same party and has been eavesdropping on our conversation, pipes up and says “I couldn’t help overhearing the last bit of your conversation about that theorem of Gauss, and I was struck by your use of the phrase ‘odd prime’. Surely you must know that the phrase is redundant; only an odd number can be prime!” The historical Nicomachus defined a prime as an odd number that can’t be expressed as a product of two smaller odd numbers, so for him, 2 wasn’t prime. My imaginary Nicomachus thinks me slow-witted for failing to notice that, even as I fault Goldbach for having an over-generous definition of the word “prime”, I myself am guilty of the same fault, by failing to notice how different 2 is from the true primes. Goldbach says that the first prime is 1, and I say that the first prime is 2, but Nicomachus says that the first prime is 3, and he takes the Law of Quadratic Reciprocity as clinching evidence: “This Law is simpler to state if the number 2 is treated separately, and you yourself have called this Law the most beautiful proposition of number theory; so you must admit that 2 isn’t truly prime!”

A lively argument ensues about the merits of our respective definitions, but it’s important to notice what isn’t at stake in this argument: the three of us agree on the facts of math, such as the uniqueness of factorization into primes or the Law of Quadratic Reciprocity. We just talk about those facts using different words. The observation that people use different words to describe a shared reality is banal when multiple languages are involved, but we somehow forget this fundamental observation about language in the context of mathematical discourse.

Definitions in math are not eternal truths. They’re human choices, shaped by our need for coherence and our desire for beauty. We as a species get to choose how we define our words. Of course, we should think hard before we define a word, and think harder before we try to uproot an established definition. But we should never forget that humans invented the words to begin with.

It’s natural for math teachers to stress precision in speech and to insist on adherence to shared conventions about the meanings of words. But an unintended consequence of teacherly fussiness can be the misimpression that the definitions of mathematical terms are revelations from on high.

This misimpression hides from students something important: although we need conventions in order to communicate, mathematical truth is deeper than mere convention. The numerical facts asserted by the Law of Quadratic Reciprocity are equally true for Christian Goldbach and Carl Friedrich Gauss and Nicomachus of Gerasa, and you and me, even if we might initially use different definitions of the word “prime” when we’re talking about them.

So, does it matter whether 1 is prime? Maybe not. But the specific way in which it doesn’t matter matters very much.

Thanks to Sandi Gubin.

This essay is a supplement to chapter 1 (“The Infinite Stairway”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds cool and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!

NOTES

#1. One way in which 1 “quacks” like a prime is the way it accords with Euclid’s Lemma, the principle that asserts that if p is a prime, then whenever the product of two integers is divisible by p, one of the two numbers or both must be divisible by p. The numbers 2, 3, 5, 7, … all have this property, and the non-prime numbers 4, 6, 8, 9, … all lack it. On which side of this dichotomy does 1 stand? Well, the proposition “Whenever the product of two integers is divisible by 1, one of the two numbers or both must be divisible by 1” is as true as “If 2+2 = 4 then 2+2 = 4” – it’s not an interesting assertion, but it’s certainly not false. So Euclid’s Lemma seems to counsel us to lump 1 together with the primes.

#2. I had a middle school classmate who’d been taught back in elementary school that “a prime is any positive integer that is divisible only by 1 and itself,” which would seem to make 1 prime, and he was taken aback in middle school to learn that, no, 1 isn’t prime after all. He felt he’d been misinformed by his earlier teachers, but our middle school teacher insisted he hadn’t been, and explained via a kind of Talmudic reasoning that the word “and” in the phrase “1 and itself” requires that the words “1” and “itself” refer to different numbers. I think that’s disingenuous; what’s more, it sets students up to make a basic conceptual mistake in algebra, namely, the mistake of thinking that when there are two or more variables sitting around, they can’t be equal to each other because “If x and y referred to the same number we wouldn’t have given them different names.” It’s best to teach kids from the start that mathematicians define a prime as a number greater than 1 with no (positive) divisors other than 1 and itself.

#3. Among the numbers of the form a + b sqrt(2), the units are the ones that satisfy a² − 2b² = ±1. This is sometimes called Pell’s equation after the 17th century mathematician John Pell, but it has interested mathematicians since the time of Pythagoras. The Indian mathematician Brahmagupta discovered a way to combine two old solutions to get a new one; for instance, by combining (a,b) = (3,2) with (a,b) = (7,5), Brahmagupta derived the solution (a,b) = (41,29). Brahmagupta probably didn’t know it, but his way of combining solutions was a disguised way of multiplying algebraic numbers: 3 + 2 sqrt(2) times 7 + 5 sqrt(2) equals 41 + 29 sqrt(2). The fruitfulness of Brahmagupta’s approach arises from the fact that the product of two units is always a unit.

#4. Our sieving procedure is making faster progress than you might think; all the composite numbers up to 25 have now been sifted out, so 2, 3, 5, 7, 11, 13, 17, 19, and 23 are all the primes up to 25.

#5. Although it’s usually credited to Gauss, the theorem was first stated and possibly proved by the Islamic mathematician Kamal al-din Al-Farisi around the year 1300.

#6: You may think that the uniqueness of prime factorizations is obvious. If so, I ask you to check that 209 × 221 equals 187 × 247, and then to tell me why you’re so sure that those four three-digit numbers aren’t prime. Or, let’s say we restrict ourselves to the number system that contains only even integers; in that restricted number system 2 × 18 equals 6 × 6, but I defy you to break down 2, 6, or 18 as a product of two even integers.

#7: If we define the special symbol (p|q) to be +1 when p has a square root in mod q arithmetic and to be −1 when p doesn’t (and to be 0 when p = q, if you’re going to be fussy), then the Law can be summarized by the equation

(p|q) (q|p) = (−1)^{(p−1)(q−1)/4}

REFERENCES

Chris K. Caldwell and Yeng Xiong, What is the Smallest Prime?

MathOverflow, “When did the career of 1 as a prime number begin and when did it end?“.

Wikipedia, Prime number.