The 3 ð ‘¥ + 1 Problem as a String Rewriting System

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