An Automated Approach to the Collatz Conjecture

Abstract

We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers. Termination of this rewriting system is equivalent to the Collatz conjecture. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses the automated method of matrix/arctic interpretations and we perform experiments where we obtain proofs of nontrivial weakenings of the Collatz conjecture. Finally, we adapt our rewriting system to show that other open problems in mathematics can also be approached as termination problems for relatively small rewriting systems. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

An Automated Approach to the Collatz Conjecture

“…Proof of Lemma 3. 16 Let T = {ρ, σ, τ } be a set of types. It is straightforward to see that, with respect to the typing…”

“…This article is an extended version of our CADE 2021 paper [16]. In addition to several small improvements throughout the article, we expanded Section 2 with extra background material to make the article more self-contained and accessible for nonexperts in rewriting.…”

A Conjecture Equivalent to the Collatz Conjecture

“…There is some recent work [4] on trying to prove the Collatz conjecture using automated termination proving techniques for rewriting systems. For this purpose, one needs a rewriting system that mimics the iterations in Collatz conjecture.…”

“…For this purpose, one needs a rewriting system that mimics the iterations in Collatz conjecture. There is one such rewriting system in [4]. The formulation presented here can also be turned into a (different) rewrite system.…”

Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3

“…• Experimental or computational method: This method uses computational optimizations to verify Collatz conjecture by checking numbers for convergence [5][6][7]. Numbers as large as 10 20 have shown no divergence from the conjecture.…”