The 3x+1 Problem as a String Rewriting System

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 twoMersenne numbers 2 −1 or their equivalents, in the sense that every integer of the form 4m 1 withm being odd is equivalent tom because both yield the same successor. II If T 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.