Why Isn't 1 a Prime Number? | Roots of Unity - Scientific American

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And how long has it been a number?, argues  Evelyn Lamb, Freelance math and science writer based in Salt Lake City, Utah.
 
An engineer friend of mine recently surprised me by saying he wasn’t sure whether the number 1 was prime or not. I was surprised because among mathematicians, 1 is universally regarded as non-prime.

Photo: Kevin Dooley Flickr (CC BY 2.0)

The confusion begins with this definition a person might give of “prime”: a prime number is a positive whole number that is only divisible by 1 and itself. The number 1 is divisible by 1, and it’s divisible by itself. But itself and 1 are not two distinct factors. Is 1 prime or not? When I write the definition of prime in an article, I try to remove that ambiguity by saying a prime number has exactly two distinct factors, 1 and itself, or that a prime is a whole number greater than 1 that is only divisible by 1 and itself. But why go to those lengths to exclude 1?
My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1⁵⁹⁴⁸²⁷×2. Excluding 1 from the primes smooths that out.

My original plan of how this article would go was that I would explain the fundamental theorem of arithmetic and be done with it. But it’s really not so hard to modify the statement of the fundamental theorem of arithmetic to address the 1 problem, and after all, my friend’s question piqued my curiosity: how did mathematicians coalesce on this definition of prime? A cursory glance around some Wikipedia pages related to number theory turns up the assertion that 1 used to be considered prime but isn’t anymore. But a paper by Chris Caldwell and Yeng Xiong shows the history of the concept is a bit more complicated. I appreciated this sentiment from the beginning of their article: “First, whether or not a number (especially unity) 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.”...

In the very most basic example, we can ask whether the number -2 is prime. The question may seem nonsensical, but it can motivate us to put into words the unique role of 1 in the whole numbers. The most unusual aspect of 1 in the whole numbers is that it has a multiplicative inverse that is also an integer. (A multiplicative inverse of the number x is a number that when multiplied by x gives 1. The number 2 has a multiplicative inverse in the set of the rational or real numbers, 1/2: 1/2×2=1, but 1/2 is not an integer.) The number 1 happens to be its own multiplicative inverse. No other positive integer has a multiplicative inverse. The property of having a multiplicative inverse is called being a unit. The number -1 is also a unit within the set of integers: again, it is its own multiplicative inverse. We don’t consider units to be either prime or composite because you can multiply them by certain other units without changing much. We can then think of the number -2 as not so different from 2; from the point of view of multiplication, -2 is just 2 times a unit. If 2 is prime, -2 should be as well.
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Source: Scientific American