Number Theory

    

An Arithmetic Reformulation of the 3x + 1 Problem Using Signed Jacobsthal Numbers

Authors: Satya Das

We establish a structural correspondence between the Collatz map and the signedJacobsthal numbers, providing an arithmetic reformulation of the 3x + 1 problem. Byrepresenting Collatz iterations through powers of signed Jacobsthal numbers, we derivenecessary and sucient conditions for the existence of cycles and for the validity ofthe coecient stopping time conjecture. This formulation translates the combinatorialdynamics of the Collatz map into explicit number-theoretic identities, revealing anunderlying algebraic framework that connects iteration, recurrence, and integrality.The results suggest a pathway toward analyzing the conj

Comments: 22 Pages. The content is changed as part of an improvement

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Submission history

[v1] 2025-11-07 01:27:01
[v2] 2025-11-23 17:36:46

Unique-IP document downloads: 330 times

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