As conciseness is one of the main mathematical features, I would like to discuss one particular instance of it. Can someone please summarize in that context the usefulness of excluding number one from the set of prime numbers? As the definition of prime numbers would be more concise without it, ie if one was included, and in fact it was at the beginning, first great contributors to number theory who laid foundations to prime number theory considered it to be prime, exclusion was introduced later, without much change in the essence of the theory, so it must have payed off somehow in terms of development of shorter expressions of consequences of somewhat longer definition, and I would like to know all places where it showed to be the case. So, the definition is, for those who are really unfamiliar with the topic: prime numbers are natural numbers that are divisible by exactly two distinct divisors, by one and by themselves. The definition that would include one, would be like this: prime numbers are natural numbers that are divisible only by one and by themselves. Note that further shortening of the definition by omitting the crucial condition “only by” would be a blunder, since all natural numbers are divisible by one and by themselves, which would of course make the definition pointless.
I know there are already many answers online to the question “why 1 is not prime”, such as https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/ for example, but I didn’t find a satisfactory one.