Yet definitions are not made at random these choices are bound by our usage of mathematics and, especially in this case, by our notation.”Ĭaldwell and Xiong start with classical Greek mathematicians. 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.
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