Numerical verification of the Collatz conjecture for billion digit random numbers

TL;DR

This work presents a practical method to numerically verify the Collatz conjecture for extremely large numbers on a standard PC by leveraging residue-class condensation and Collatz polynomials. It introduces a polynomial framework, $C_{k,r}$, that encodes $T^{(k)}$ on residue classes modulo $2^k$, enabling condensation of multiple iterations into single steps via a triple $(a,b,c)$ representation and fast evaluation. The paper provides an implementation strategy using binary splitting and Magma, and reports empirical success verifying random numbers with up to $10^{10}$ decimal places, observing a stopping-time distribution near normal with mean about $\log(n_0)/\log(2/\sqrt{3})$. While the results demonstrate practical scalability and efficiency, they do not constitute a proof of the conjecture; instead, they illustrate the capabilities of modern computer algebra systems for exploring large instances of the problem.

Abstract

The Collatz conjecture, also known as the 3n+1 problem, is one of the most popular open problems in number theory. In this note, an algorithm for the verification of the Collatz conjecture is presented that works on a standard PC for numbers with up to ten billion decimal places.