Why are the prime numbers distributed loggishly? a very nice heuristic argument

A nice group theory proof. I think it's a theorem by Galois. It was a homework exercise in first year algebra. The professor gave a very short proof using a minimality argument, and said he'd be intrigued if anyone managed to prove it without this sleight of hand. But I did! I don't remember the very short proof, unfortunately. Maybe I'll dream hard and dream it up.

A very nice topology proof. I like it
because
of an interesting lemma I proved along the way: if U is a connected
open
set in R^{n} and the boundary of U isn't connected, then the
complement of U can't be connected. Also it has an interesting
perspective on homotopies.

There are lots and lots of subfields of C that are isomorphic to R. Including some that intersect R only in the algebraic numbers.

No groups of order 720 are simple.
Showing groups aren't simple is a kind of game: how can you see (or smell)
your way through a numerological thicket? 720 is a
hard number to prove this for, because A_{6} has order 360 and it's
simple. So there's a big almost counter-example sitting like a large
invisible lump in the logical thicket. You have to calculate your way
around it without knowing where it is. I wanted to see if
I could do it. I kept on posting proofs on usenet, and it was very
embarrassing, a group theorist would find an error.
Math is
a language you can talk and logic is the grammar of it. When I was
studying math in college I'd make mistakes right after I came back from
vacation, then I'd remember how to think in math and stop
being wrong.

a program to triangulate compact connected 2-manifolds