Is 1 Prime, and Does It Matter?

Summary

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 a...