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.
But a paper by Chris Caldwell and Yeng Xiong shows the history of the concept is a bit more complicated. 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. My original plan of how this article would go was that I would explain the fundamental theorem of arithmetic and be done with it. Excluding 1 from the primes smooths that out. If 1 were prime, we would lose that uniqueness. 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. But why go to those lengths to exclude 1? 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 itself and 1 are not two distinct factors. The number 1 is divisible by 1, and it’s divisible by itself. 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.
I was surprised because among mathematicians, 1 is universally regarded as non-prime. An engineer friend of mine recently surprised me by saying he wasn’t sure whether the number 1 was prime or not.