Convergence of some perturbed sequences of rational powers and application to syracuse problem

TL;DR

This paper introduces 'Branch sequences' of perturbed rational powers and proves their convergence under certain conditions, applying this to confirm the Collatz conjecture, a longstanding problem in number theory.

Contribution

It establishes convergence of a new family of perturbed rational power sequences and applies this to prove the Collatz conjecture.

Findings

Convergence of 'Branch sequences' under controlled perturbations.

Syracuse sequences are shown to be 'Branch sequences' meeting convergence conditions.

Confirmation of the Collatz conjecture based on sequence analysis.

Abstract

Sequences of rational powers \left( \xi\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse, Z-number and waring problems. Conjectures from such problems are known to be intractable and only few partial results exist until now. In this paper, we study a family of perturbed sequences of rational powers called 'Branch sequences' of the form \left( S_{n}=\left( \xi+\Sigma_{n} \right)\left( \frac{p^{n}}{q^{n+e_{n}}} \right) \right)_{n\ge 0}. Under the assumption that such sequences are deterministic and they have controlled positive perturbations, we establish the convergence result: min_{n\ge 0}(S_{n})\le q^{2}. As an application, we show that Syracuse sequences are 'Branch sequences' with all the required conditions for convergence and therefore this confirms the Collatz conjecture. Keywords: Sequences of rational powers, Syracuse conjecture, Collatz problem, 3x+1 problem.