Two remarks on the Collatz cycle conjecture

TL;DR

This paper provides a concise proof related to bounds on certain cycles in the Collatz problem and reformulates the conjecture as a purely arithmetic problem, offering new perspectives on this longstanding mathematical challenge.

Contribution

It offers a short proof of bounds on perigees of specific Collatz cycles and rephrases the conjecture as an arithmetic problem, advancing understanding of the problem's structure.

Findings

Bounded perigees of $(3x+d)$-cycles established

Collatz cycle conjecture reformulated as an arithmetic problem

Provides new insights into the structure of Collatz cycles

Abstract

We give a short proof of Belaga's result on bounds to perigees of $(3x+d)$-cycles of a given oddlength. We also reformulate the Collatz cycle conjecture which is rather a algorithmic problem into a purely arithmetic problem.