The 3x + 1 Problem as a String Rewriting System

Abstract

The 3x + 1 problem can be viewed, starting with the binary form for any n ∈ N, as a string of “runs” of 1s and 0s, using methodology introduced by Błażewicz and Pettorossi in 1983. A simple system of two unary operators rewrites the length of each run, so that each new string represents the next odd integer on the 3x + 1 path. This approach enables the conjecture to be recast as two assertions. (I) Every odd n ∈ N lies on a distinct 3x + 1 trajectory between two Mersenne numbers (2k − 1) or their equivalents, in the sense that every integer of the form (4m + 1) with m being odd is equivalent to m because both yield the same successor. (II) If Tr(2k − 1)→(2l − 1) for any r, k, l > 0, l < k; that is, the 3x + 1 function expressed as a map of k′s is monotonically decreasing, thereby ensuring that the function terminates for every integer.