$7x\pm1$: Close Relative of Collatz Problem

TL;DR

This paper introduces a new iterated function related to the Collatz problem, demonstrating complex oscillations and growth, and conjectures all positive integers eventually reach 1, supported by heuristics and computational evidence.

Contribution

It proposes a novel function similar to Collatz, providing heuristic and computational support for the conjecture that all positive integers return to 1.

Findings

Iterates oscillate wildly and grow rapidly

Heuristic argument supports the conjecture

Computational results back the conjecture

Abstract

We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. We conjecture that the orbit of arbitrary positive integer always returns to 1, as in the case of Collatz function. The conjecture is supported by a heuristic argument and computational results.