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 eπ√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.
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.
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.
