Consecutive Integers and the Collatz Conjecture

TL;DR

This paper investigates the behavior of consecutive integers under the Collatz conjecture, revealing an infinite family of counterexamples to a previously proposed conjectural family of conditions.

Contribution

It provides the first known infinite family of counterexamples to Garner's conjecture on Collatz heights for consecutive integers.

Findings

Identifies an infinite family of counterexamples to Garner's conjecture.

Shows that pairs of consecutive integers can have the same Collatz height more frequently than previously thought.

Abstract

Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs. Our main result is an infinite family of counterexamples to Garner's conjecture.