Copyright © 2010 Joseph Sinyor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The 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 path. This approach enables the conjecture to be recast as two assertions. (I) Every odd lies on a distinct trajectory between two Mersenne numbers () or their equivalents, in the sense that every integer of the form () with being odd is equivalent to because both yield the same successor. (II) If for any , ; that is, the function expressed as a map of 's is monotonically decreasing, thereby ensuring that the function terminates for every integer.