Half an Induction Proof of the Collatz Conjecture

Abstract

Summary In a straightforward proof by induction of the Collatz conjecture, it is easy to show that the statement is true for n = 1 (the base case) but proving the induction step is more challenging—in fact, no one has done so to date. In this work, a different induction proof of the conjecture, based on a binary representation of the starting integer, is given in which it is possible to prove the induction step but there is no known proof for the base case. In fact, if one could prove the base case, then the Collatz conjecture is true.