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. Plato’s nephew Speussippus, for instance, 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, admitted that “the number 1 is certainly not composite in the same sense as the number 6,” but maintained that “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 number and a unit.

AVOIDING AWKWARDNESS

I suspect one reason I suspect 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. Not because of any one thing, but because of dozens of different ways in which treating 1 as