# Convergence verification of the Collatz problem

@article{Barina2020ConvergenceVO, title={Convergence verification of the Collatz problem}, author={David Barina}, journal={The Journal of Supercomputing}, year={2020}, pages={1-8} }

This article presents a new algorithmic approach for computational convergence verification of the Collatz problem. The main contribution of the paper is the replacement of huge precomputed tables containing $$O(2^N)$$ O ( 2 N ) entries with small lookup tables comprising just O ( N ) elements. Our single-threaded CPU implementation can verify $$4.2 \times 10^9$$ 4.2 × 10 9 128-bit numbers per second on Intel Xeon Gold 5218 CPU computer, and our parallel OpenCL implementation reaches the speed…

## 23 Citations

### There are no Collatz-m-Cycles with $m\leq 90$

- Mathematics
- 2022

The Collatz Conjecture or Syracuse Problem considers recursively defined sequences of positive integers where n is succeeded by n2 , if n is even, or 3n+1 2 , if n is odd. The conjecture states that…

### An Algorithm for Linearizing the Collatz Convergence

- Mathematics, Computer ScienceMathematics
- 2021

An ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that it is proved apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number.

### An Automated Approach to the Collatz Conjecture

- Mathematics, Computer ScienceCADE
- 2021

A rewriting system is constructed that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers and it is shown that other open problems in mathematics can also be approached as termination problems for relatively small rewriting systems.

### Orbits Theory. A Complete Proof of the Collatz Conjecture

- Mathematics
- 2021

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be…

### How to Claim a Computational Feat

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2021

This article describes a non-interactive protocol in which the prover provides (a digest of) the computational trace resulting from processing x, for randomly chosen x ∈ U .

### lrsarith: a small fixed/hybrid arithmetic C library

- Computer ScienceArXiv
- 2021

We describe lrsarith which is a small fixed precision and hybrid arithmetic C library for integers and rationals that we developed for use in the lrslib library for polyhedral computation. Using a…

### The Collatz conjecture and the quantum mechanical harmonic oscillator

- MathematicsJournal of Mathematical Chemistry
- 2021

By establishing a dictionary between the QM harmonic oscillator and the Collatz process, it reveals very important clues as to why the Collatz conjecture most likely is true. The dictionary requires…

### The Collatz tree as an automorphism graph: a cotree density proof of the $3x + 1$ conjecture

- Mathematics
- 2020

The Collatz conjecture maintains that each natural number n is a node in the Collatz tree with a root path to 1 generated by iterations of the Collatz function, which connects even n to n/ 2 and odd…

### Observations on cycles in a variant of the Collatz Graph

- Mathematics, Computer Science
- 2021

This paper explores a variant of the Collatz Graph, which allows the 3x+1 operation to be applied to both even and odd integers, and proves an analogous function for this variant, called the Loosened Collatz Function (LCF), and observes various properties of the LCF in relation to tuples and outputs.

### Is the Syracuse falling time bounded by 12?

- Mathematics
- 2021

Let T : N → N denote the 3x + 1 function, where T (n) = n/2 if n is even, T (n) = (3n + 1)/2 if n is odd. As an accelerated version of T , we define a jump at n ≥ 1 by jp(n) = T (l)(n), where l is…

## References

SHOWING 1-10 OF 13 REFERENCES

### GPU-accelerated Exhaustive Verification of the Collatz Conjecture

- Mathematics, Computer ScienceInt. J. Netw. Comput.
- 2017

This paper presents an implementation that performs the exhaustive search to verify the Collatz conjecture using a GPU and achieves a speed-up factor of 249 over the sequential CPU implementation.

### The Ultimate Challenge: The 3x+1 Problem

- Mathematics
- 2011

This book reports on what is known on the problem of the 3x 1 problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computation.

### Maximum excursion and stopping time record-holders for the problem: Computational results

- MathematicsMath. Comput.
- 1999

This paper presents some results concerning the search for initial values to the so-called 3x+1 problem which give rise either to function iterates that attain a maximum value higher than all…

### The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)

- Mathematics
- 2006

The 3x+1 problem concerns iteration of the map T(n) =(3n+1)/2 if n odd; n/2 if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the forward orbit of n includes the integer 1.…

### The 3x+1 problem: An annotated bibliography (1963--1999)

- Mathematics
- 2003

The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the…

### On an Ulam's problem

- MathematicsANNALI DELL UNIVERSITA DI FERRARA
- 1996

SuntoIn questo lavoro si dimostrano l'esistenza e l'unità di un flusso vettorialeX in ℝn,n≥3, che ha le seguenti proprietà:
(1)X non ha punti stazionari in ℝn;(2)tutte le orbite diX sono…

### The 3x+1 problem: An annotated bibliography

- Medicine
- 2003

Psoriasis can be effectively arrested and controlled by the administration of from about 50 milligrams dosage to about 150 milligrams dosage per day of pentazocine.