[feoren]

All models are wrong, but some models are useful. It's not useful to consider 1 prime, so we don't. You're free to invent a new model of math where 1 is prime and see where it takes you; nobody will be offended. This happens all the time: "but what if we could take the square root of a negative number? What then?", etc. 99% of the time, this leads to a theory that is provably inconsistent and therefore useless. Out of the remaining 1%, about 99% of the time it leads to a mathematics that is simply less useful than what we have now. So it goes with making 1 prime. Out of the remaining cases, about 99% of those turn out to be identical to an already existing mathematical theory, which is interesting (and possibly publishable), but not hugely useful. But about 1% of 1% of 1% of the time, these exercises result in actual new math that can tell us new things about reality and solve problems we couldn't solve before.

This is not one of those times.

[rtkwe]

I've always wondered what actually breaks if 1 is prime or conversely what defining 1 as not prime gives us. Got just far enough into my math degree before switching to CompSci to stay of of universities the rest of my life to want to know.

[feoren]

The biggest problem is that you lose unique prime factorization. With prime factorization, I get a unique representation of every positive integer. Let's consider a way to write positive integers in "base prime", similar to base 10 or base 2. I'll start counting from 1 and write numbers as a tuple of prime factors. Similar to base 10, "base prime" has an infinite set of 0s that we're leaving out for brevity (e.g. 19 = 0000019), although it's on the right side instead of the left.

    1 = () = (0, 0, 0, 0, 0, ...)
    2 = (1) = (1, 0, 0, 0, 0, ...)
    3 = (0, 1)
    4 = (2)
    5 = (0, 0, 1)
    6 = (1, 1)
    7 = (0, 0, 0, 1)
    8 = (3)
    9 = (0, 2)
    10 = (1, 0, 1)

The i th position in every tuple is the power of the i th prime in the factorization of that number. So 10 = (1, 0, 1) = 2^1 * 3^0 * 5^1. 84 would be (2, 1, 0, 1) = 2^2 * 3^1 * 5^0 * 7^1. If we have unique factorization, there is exactly one way to write every positive integer like this, and there are many insights we can gain from this factorization. If 1 is prime, then we can write 6 = 1^257 * 2^1 * 3^1, or any other power of 1 we like. We just gain nothing from it.

There are often many equivalent ways to define any mathematical object, and I'm sure there are plenty of ways to define a prime number other than "its only factors are itself and 1". These other definitions are likely to obviously exclude 1. One obvious one is the set of basis coordinates in this "unique factorization" space that I just laid out here. And we're never really excluding or making a special case for 1, because 1's factorization is simply the absence of any powers -- empty set, all 0s, whatever you want to call it.

Keep in mind that "unique factorization" turns out to be very interesting in all sorts of other mathematical objects: rings, polynomials, symmetries, vector spaces, etc. They often have their own notion of "prime" or "primitive" objects and the correspondence with integer-primes is much cleaner if we don't consider 1 prime.

[samf]

Some examples are in these comments, e.g. the Fundamental Theorem of Arithmetic. The Sieve of Eratosthenes is an amusing outcome, where 1 is the only prime if you take it literally.

But also mentioned elsewhere in the thread: if we declared 1 to be a prime, then many (I daresay "most") of our theorems would have to change "prime number" to "prime number greater than one".

[dullcrisp]

If you defined 1 to be a prime but to not be odd then some theorems could stay the same.

[samf]

Ha, yes, I was thinking of the theorems that refer to "odd prime" to exclude 2. :-)

[samf]

This is the best answer.

We could declare 4 to be a prime number, and keep the rest of the definition the same. Instead of just saying "no", you could ask, "okay, what would that do for us?" If there isn't a good answer, then what's the point? And usually, you're not in the 1% of 1% of 1%.