What do the numbers 2 and 7 have in common? It's hard to get a really good answer.
I asked a LLM and it said "they're the only prime factors of 14". 😆 Fun to analyze why that's a bad answer, though true.
Better: 2 and 7 are both Heegner numbers. Heegner numbers are square-free natural numbers n such that unique factorization holds for algebraic integers in ℚ√(-n). The only Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, and 163. This is related to famous facts like
exp(π√43) ≈ 960³ +744 - 0.00022
exp(π√67) ≈ 5280³ + 744 - 0.0000013
exp(π√163) ≈ 640320³ + 744 - 0.00000000000075
But for 2 and 7, I don't think we get anything quite so exciting.
Even better: 2 and 7 are primes for which X₀(p) has genus zero. X₀(p) is defined to be the compactification of the moduli space of elliptic curves equipped with a cyclic subgroup of order p. This has genus zero precisely when p = 2, 3, 5, 7, and 13.
So, if we say "2 and 7 are Heegner numbers that are primes for which X₀(p) has genus zero" then we're really saying something, since the only other member of this elite club is the number 3.
But I don't know number theory to have this leap out at me like it would for some mathematicians I know. John McKay - the Monstrous Moonshine guy - would have gotten it really quick.
(2/n, n = 2)
https://en.wikipedia.org/wiki/Heegner_number
