At the risk of repeating what I wrote here years ago: What finally got me interested in algebra and number theory as I approached my 50th birthday is the theorem (conjectured by Gauss, proved by Heegner, reproved by Stark) -- that among imaginary quadratic fields Q(sqrt(-d), their rings of algebraic integers are unique factorization domains precisely for 9 of these fields, and these 9 rings can be expressed as Z[sqrt(-d)] for d = 1,2,3,7,11,19,43,67,163. (Gauss already knew that these were UFD's, and conjectured that there were no others; that's what Heegner proved in 1952; see < http://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem#History >.) As may have been mentioned here, Conway & Guy give a good heuristic idea of how the proof proceeds in their book Numbers (thanks again for that, Rich & Hilarie). (I love theorems with a strange set of exceptions like that. (E.g., the automorphism group Aut(S_n) of the symmetric group S_n is isomorphic to S_n itself for precisely all n not equal to 2 or 6.) --Dan Hans writes: << I'm going to guess that the "positive" Gaussian quadrant illustrated by Conway and Guy can be rotated into the *top-right* quadrant without loss of the unique-factorization principle.
Sometimes the brain has a mind of its own.