Agree. Almost all theorems you’d need to exclude 1, there’s a bunch that exclude 2, a notable number that exclude 2 and 3, and a rapidly declining number that exclude primes up to larger numbers.
Which reminds me of the weird fact - pick any positive integer and factorize it. Sort the prime factors by magnitude. The median value for the second largest prime factor is 37.
And for some of them, like the one that’s called The Prime Number Theorem, it doesn’t matter whether you include 1, 2, or 3, because it’s about the asymptotic density of primes.
Is there a generalization of this to the largest prime factor, third-largest, etc.? And how does one account for the fact that some numbers don’t even have a second-largest prime factor?
Oops. I said that wrong. 37 is the median value for the prime that’s second smallest in the factorization. For 6, 30, 42, etc, 3 is the second smallest prime (2 is the smallest, and there can be any number of larger primes in the factorization). For a prime number, there is no second prime in the factorization.
Here’s a link to a discussion of how to calculate that 37 is the right number.
I’m not sure if there’s a generalization
Thanks. I understand it now.
Is there anyone really all that inconvenienced by the current definition excluding one? There are probably thousands and thousands of people confused by the current definition and many of those would argue that it should include one but do any of those people actually use prime numbers for anything?
'Zactly.
Useful for pros is useful. Useful for spectators is fanservice.
But it always irks me when people make a big deal about 2 being the only even prime. That just means it’s the only prime divisible by 2. Big whoop. Every prime number is the only prime number divisible by itself.
You’re right - if humans had a word for dumisible by three, it would be just as silly to be surprised that 3 was the only prime that was three-ish
Simplest answer is that 1 is a unit and units are neither prime nor composite. The problem is that units are never talked about. What is a unit? if a X b = 1 then a and b are units.
I like that we now have a word for dumisble. 
Doh! Divisible
Granted.
But what’s interesting to me about that is why we make a deal out of even/odd = divisible/indivisible by 2. But don’t have magic words for the others. Why is divisble by 2 considered interesting enough to have a name, but by 3 or 5 or … isn’t?
Hellifino.
Even/odd appears to go way back. People would sort objects into pairs to make sure they evened out. An extra one would be the odd one out. But the extra also could break a tie and settle a vote. Numbers far preceded mathematics and even arithmetic so they had time to develop personalities of their own.
Evens and odds made their way into superstitions, in which even numbers had a set of traits (" Old English efen “level,” also “equal, like; calm, harmonious; equally; quite, fully; namely,”) and odd numbers had a different set. Over time, individual numbers took over. Four is unlucky in China but 13 is unlucky in the West. Seven has been a mystical number for thousands of years.
I’ll leave it to the mathematicians here to speak to how important even/odd is to formal work, but I’ll bet that schoolkids still get taught to pair things very early on.
It is worth pointing out that 2 and −2 are prime numbers, while 1 (and −1) are units, which one generally wants not to be prime or composite; 1 is an empty product, as you say.
We cannot expect a unique word describing divisibility by every possible number, can we?
Those kinds of properties are considered interesting enough in numerology, arithmancy, gematria, isopsephy, etc., but I am not sufficiently versed in the esoteric literature.
There’s no reason there couldn’t be a productive suffix or something like that which would allow such a thing. Like “threeven” for divisible by 3, “fourven” for divisible by 4, etc. I wonder if any language has something like that.
I would not expect a distinct word for each divisibility. Just that for numbers small enough to be used in everyday counting and grouping. Where 3 and 5 might have a decent claim on being as useful as 2. OTOH 23, or 37, or some arbitrary 6 digit number? Not so much. In fact, not at all.
I think @Exapno_Mapcase probably nailed it with his historical perspective. Language about counting and grouping grew up long before arithmetic did, much less mathematics and number theory. Lots of things naturally come in twos.
In societies that used 5-based tally marks to record counts of [whatever], I could readily imagine them having a word that meant to them “several complete 5-tallies with no excess items”, but which from our math-centric POV means “divisible by 5”.
A prime factor yields another factor. A composite number Xc divided by one of its prime factors Pf is Xc / Pf = xf , which is either a prime or composite factor of Xc . If Pf is a prime number, xf is a different, smaller number than Xc . Hence, 1 is not a prime number because, if Pf = 1, xf will equal Xc . A prime number is one which cannot be factored into a smaller integer, but 1 is not prime because it cannot be a unique factor of any composite number.
Well, as a grade school student, I was taught that prime numbers have only two factors, one and itself. Example: 7 is a prime number because it has just two factors, 1 and 7. One wasn’t a prime number because it has only one factor because one IS itself.