On a general Syracuse problem with conjectures

TL;DR

This paper investigates a generalized version of the Syracuse problem, establishing necessary conditions for non-trivial cycles, exploring properties via linear logarithmic forms, and proposing new conjectures with illustrative examples.

Contribution

It introduces new conjectures and necessary conditions for the Syracuse problem's cycles, expanding understanding of its complex behavior.

Findings

Necessary conditions for non-trivial cycles identified

Properties based on linear logarithmic forms established

New conjectures proposed and illustrated with examples

Abstract

In this paper, we study a general Syracuse problem. We give some necessary conditions concerning the existence of eventual non trivial cycles. Some properties based on linear logarithmic forms are established. New general conjectures are given. To illustrate the behavior of such a problem, some particular examples are presented.