# A polynomial approach to the Collatz conjecture

## TL;DR

This paper investigates the Collatz conjecture through polynomial representations derived from binary systems, showing that polynomial degrees tend to decrease, offering a weak inductive proof towards the conjecture.

## Contribution

Introduces a polynomial approach based on binary systems to analyze the Collatz conjecture, providing a novel perspective and a weak proof outline.

## Key findings

- Polynomial degrees decrease on average after finite steps
- Provides a weak inductive proof of the conjecture
- Offers a new algebraic framework for Collatz analysis

## Abstract

The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture by using induction with respect to the degree of the polynomials.