A Solution to the 3x + 1 Problem

We present several proofs of the 3x + 1 Conjecture, which asserts that repeated iterations of the function C(x) = (3x + 1)/(2a) always terminate in 1 Here x is an odd, positive integer, and a is the largest positive integer such that the denominator divides the numerator. Our first proofs are based on a structure called “tuple-sets” that represents the 3x + 1 function in the “forward” (as opposed to the inverse) direction. In “Most Recent Proof of the Conjecture” on page 11, we show that, because the “number”1 of tuples in each tuple-set is the same, regardless if counterexamples exist or not, and because the set of all non-counterexamples is the same, regardless if counterexampels exist or not, it follows that counterexamples do not exist. In our next two proofs, we show, by a simple inductive argument, that the contents of the set of all tuple-sets is the same, regardless if counterexamples exist or not, and from this we conclude that counterexamples do not exist. “Third Proof” is based on a structure called “recursive ‘spiral’s” that represents the 3x + 1 function in the inverse direction. We show that, because a large number of consecutive odd, positive integers are known, by computer test, to be non-counterexamples, it follows, by an inductive argument based on certain fundamental properties of recursive “spiral”s, that the set of all tuples in each infinite set of recursive “spiral”s is the same regardless if counterexamples exist or not. We infer from this that counterexamples do not exist. As far as we have been able to determine, our approach to a solution of the Problem is original. 1. This is an abuse of language that it made precise in step 2 of the proof.