h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I made significant progress on proving part of the 3n+1 conjecture today (the part that says no sub-sequence other than {4, 2, 1} repeats). A major theorem (not proved yet) is; The limbs ending in an element of S1 are either dead, one-element limbs, two-element limbs, or four-element limbs. (See the Guide Entry "An Approach to the 3n+1 Problem" for the definition of these terms.) An elaborate induction argument would be required to prove this. I'm just now proving some rudimentary theorems about least-residue trees. Much work remains to be done, but it's clear that there's nothing profound or unprovable about this part of the 3n+1 conjecture.