The Problem as a String Rewriting System

The 3+1 problem can be viewed, starting with the binary form for any ∈, 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 3+1 path. This approach enables the conjecture to be recast as two assertions. (I) Every odd ∈ lies on a distinct 3+1 trajectory between two Mersenne numbers (2−1) or their equivalents, in the sense that every integer of the form (4+1) with being odd is equivalent to because both yield the same successor. (II) If (2−1)→(2−1) for any ,,>0, <; that is, the 3+1 function expressed as a map of 's is monotonically decreasing, thereby ensuring that the function terminates for every integer