On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations

TL;DR

This paper explores the deep connection between even binary palindromic words and the Collatz-Hailstone iterations, offering new reformulations and an analytic expression for the problem.

Contribution

It introduces a novel analytic expression for the Collatz problem using trailing zeros sequences and links it to binary palindromes, providing new reformulations.

Findings

Derived a single branch formula with a unique fixed point

Connected the Collatz problem to the discrete derivative of fixed points of a reflection operator

Presented multiple equivalent reformulations of the Collatz problem

Abstract

The celebrated $3x+1$ problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula $f(x)+1$ with a unique fixed point. The resultant formula $f(x)$ is also found to coincide with that of the discrete derivative of the sorted sequence of fixed points of the reflection operator on even binary palindromes of fixed even length \textit{2k} in any interval $[0\cdots2^{2k}-1]$. A set of equivalent reformulations of the problem are also presented.