3x+1 Minus the +

Kenneth G Monks

doi:10.46298/dmtcs.297

Abstract

We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.

Highlights

Introduction and Main ResultsThe famous 3x + 1 conjecture states that for every n ∈ Z+ there exists k ∈ Z+ such that T k (n) = 1 where T (n) =1 2 n if n is even 3 2 n + 1 2 if n is odd. and Tk = T ◦T
A Fractran program consists of a finite list of positive rational numbers, [r1, . . . rt ]
The state of a Fractran machine consists of a single positive integer

Summary

Introduction and Main Results

The famous 3x + 1 conjecture (cf [3],[4]) states that for every n ∈ Z+ there exists k ∈ Z+ such that T k (n) = 1 where. ◦···◦T denotes the k-fold composition of with itself

If we let

Compare this somewhat unwieldy expression with the much simpler one

The Fractran program

This yields the system of linear equations