Convergence and Divergence of Mersenne Variations of the $3x+1$ Function

TL;DR

This paper explores variations of the Collatz problem involving Mersenne numbers, analyzes their convergence properties, and proves a key theorem about the absence of divergent trajectories in these functions.

Contribution

It introduces Mersenne-based variations of the Collatz function, studies their convergence behavior, and provides a proof for the No Divergent Trajectories Theorem.

Findings

Identified convergent cycles in Mersenne variations

Proved the No Divergent Trajectories Theorem for these functions

Showed why similar proofs do not extend to other functions

Abstract

The Collatz problem is one of many names (the Collatz Problem, the Syracuse Problem, the Hailstone Problem, the 3x+1 problem). Most commonly, however, the problem goes by either the 3x+1 problem or the Collatz problem. In addition to having many names, the Collatz problem has many variations, such as those in the form introduced by Jeffrey Lagarias in 1985. This writing discusses several variations of the Collatz function which involve the Mersenne numbers. Following that, we observe the convergent cycles of these functions which we can then relate back to the original Collatz 3x+1 function. Lastly, we give a proof of the No Divergent Trajectories Theorem and show why the same cannot be shown for similar functions.