The 3x+1 Problem on Rational Numbers by Rachel Schmidt '25

The 3x+1 Problem on Rational Numbers by Rachel Schmidt ’25, Friday May 9, 1:00 – 1:50pm, North Science Building 015, Wachenheim, Mathematics Thesis Defense

The Collatz conjecture famously hypothesizes that, for an odd number x, iteratively applying the map C(x) = (3x+1)/2^k will always lead to containment in the set {1, 2, 4}. We extend the Collatz conjecture (also known as the 3x+1 problem) to non-integer rational numbers through a modified mapping function. We present an analog conjecture for the rational Collatz variant and provide evidence for its plausibility. In particular, we introduce the concept of a Collatz tree and provide a complete characterization of the structure of such trees.