There are tons of them! For example, the class of numbers n such that there is a Platonic solid made of n-gons. This class only has the numbers 3, 4, and 5. You can get other examples any time there is an interesting mathematical structure with only finitely many examples.
Well, yes, obviously. I was hoping for something number-theoretic, though. Let me reword the title.
Haha, ok, how about numbers n such that there are nontrivial solutions to a^n + b^n = c^n
My point is that interesting (non-)existence results give examples of the type I thought you were asking for.
Oh yeah, Fermat's Last Theorem. I bet I would have thought of that right away if I was a bit older. The Wiefrich primes came up elsewhere here, and they have a kind of similar background.
Thanks for the answers!
I don't know if this is what you mean by classes, but the online encyclopedia of inter sequences has a lot of interesting finite sets. Maybe you mean something else though?
That's actually a philosophy or psychology question, but mathematicians talk about interesingness a lot.
I didn't want someone to reply "1 2 3 4", basically. It has to actually be significant somehow.
so couldn’t any set become interesting (in the common sense) just as soon as someone becomes interested? e.g. by developing a theorem that incorporates it? it doesn’t feel like it adds anything to qualify an expression as mathematically interesting because it’s the creativity of the mathematician that makes the interest. maybe it’s just a social ask: “have it be interesting to you, or else keep it to yourself”, heh.
Philosophy is fun too!
so couldn’t any set become interesting (in the common sense) just as soon as someone becomes interested? e.g. by developing a theorem that incorporates it?
Yes, I think so, especially if the theorem is itself profound somehow. It doesn't even have to be in a theorem, if it has a really simple definition that non-obviously leads to a finite set that would be enough for me in this question.
it doesn’t feel like it adds anything to qualify an expression as mathematically interesting because it’s the creativity of the mathematician that makes the interest.
I'm reminded of an argument I heard once that there are no uninteresting positive reals, because being the smallest such number would itself be interesting. That seems faulty to me, it just means it's a set with no minimal element, which can exist even in an interval. The infimum would have to be 0.
maybe it’s just a social ask: “have it be interesting to you, or else keep it to yourself”, heh.
Stepping into linguistics for a moment, have you heard of pragmatics? Not sharing irrelevant information is a common unwritten rule in conversations. I specified it here because someone might incorrectly assume I'm unaware you can build arbitrary sets of natural numbers.
The Heegner Numbers. These are the n such that ℚ[√-n] has unique factorisation. There are exactly 9 of them:
1, 2, 3, 7, 11, 19, 43, 67, 163.
A famous fact about them is that 163 being a Heegner Number leads to e^(π√163) being very close to a whole number.
262537412640768743.99999999999925…
TIL about prime-generating quadratic polynomials, as well. I feel like I'm destined to use one in code now. The logic behind e^π√163^ looks like more than I can absorb today, haha.
Because I find Wikipedia doesn't explain it in the best way, a quadratic field like ℚ[√-n] is literally just the field of rationals with √-n and all the new numbers you can make with it added.
Thanks!
TL;DW for other people, the Wiefrich primes which are currently (and maybe actually) just 1093 and 3511. There's other interesting infinite sequences if you do watch, as well as one lemon.
The number of platonic solids in different dimensions could be interesting. https://math.ucr.edu/home/baez/platonic.html
Hey future people. Just necroing this to point out the best answer, which we missed. Fermat primes, of which the highest known is currently 65537. It's a very interesting set, because they determine which shapes can be constructed by compass and straightedge, which might be the oldest big question in recorded mathematics.
Like the Wiefrich primes, it's possible there's more, and unlike them it's not, as far as I can tell, widely thought they are finite (it's more up in the air). However, I think the interestingness outweighs that.