https://doi.org/10.1140/epjs/s11734-026-02343-6
Regular Article
Polar recurrence plots on Collatz sequences devoid of exact recurrences
Loyola University Chicago, Health Sciences Campus, 2160 South First Avenue, 60153, Maywood, IL, USA
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Received:
29
January
2026
Accepted:
21
April
2026
Published online:
19
May
2026
Abstract
The Collatz conjecture has long attracted mathematical interest because its simple rules give rise to complicated patterns resembling physical processes such as diffusion, random walks, dispersion of gases, and chaotic turbulences, to name a few. Collatz trajectories are torturous, large in magnitude, yet rule driven, with sensitive dependence on initial conditions. Still, all trajectories terminate with the ubiquitous 3-point tail of 4–2–1. Within any one trajectory, independent of length, no scalar values ever repeat. This means that Collatz processes can be categorized as terminal dynamics with zero recurrences. This study was designed to explore patterns within Collatz sequences in the polar recurrence space from both qualitative and quantitative perspectives. Collatz seed values (initial points) ranged from 2 to 1,048,576 (220) and generated 1,048,575 unique trajectories. Random shuffling of this large set was detected in 6 or 12 quantifiers extracted from the plots. The total data set was partitioned into 4 groups depending upon the seed integers: total set; prime numbers, anti-prime numbers, even numbers, odd numbers. It came as no surprise that none of the 12 sensitive quantitative markers were able to show any differences between trajectories seeded by prime versus anti-prime integers, or seeded by odd versus even integers. It is concluded that, from the polar recurrence perspective, Collatz sequences are rule-based systems that give rise to random-like processes. This study confirms other methodologies that have come to the same conclusion. It is mused that Collatz systems may be related to stochastic resonance, among many other possibilities, where definitive structures arise out of random noise.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
