What question did this study set out to answer?

This work aims to explore the termination behavior of the Collatz-type map T3,5(n) and its convergence properties.

May 12, 2026Open Access

A GENERALIZED COLLATZ MULTI-CYCLE TERMINATION EXAMPLE: THE 3x+5 MAP AND ITS SOURCE DECOMPOSITION

Key Points

This work aims to explore the termination behavior of the Collatz-type map T3,5(n) and its convergence properties.
Analyzed the Collatz map T3,5(n) with a focus on even and odd sequences.
Utilized source decomposition through an anti-9 lifting approach.
Developed certificates ensuring termination across identified trajectories.
Demonstrated that every trajectory of the map converges to one of six known positive terminal cycles: 1, 5, 19, 23, 187, and 347.
Established a well-founded descent associated with the source parameter u.
Formulated a terminating certificate forest with specific roots that validate convergence.

Abstract

Abstract. We study the Collatz-type mapT3, 5 (n) = n/2, n even, 3n +5, n odd. We formalize a conditional convergence strategy based on: (i) a reduction to sources 3u (with u odd) via an anti-9 lifting, (ii) a component-wise well-founded descent on the sourceparameter u produced by dyadic leaf certificates modulo Mt = 9 · 2L+6t (with L = 3), and (iii) a terminating certificate forest with six roots U0 = 1, 3, 5, 17, 41, 57. Under these hypotheses, every trajectory reaches one of the six empirically known positive terminal cycles (seed values1, 5, 19, 23, 187, 347)

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