On two conjectures for M&m sequences

TL;DR

This paper investigates the complex dynamics of M&m sequences and confirms two conjectures for certain initial conditions, advancing understanding of their behavior.

Contribution

It proves two conjectures related to M&m sequences for a subset of initial conditions, expanding theoretical knowledge.

Findings

Two conjectures are proven true for specific initial conditions.

M&m sequences exhibit complex but analyzable dynamics.

The study advances understanding of mean-median map behavior.

Abstract

In this paper, the recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original. This process, although seemingly simple, gives rise to complicated dynamics. The main result is that two conjectures put forward by Chamberland and Martelli are shown to be true for a subset of possible starting conditions.