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Cyclic Arithmetic and Combinatorial Operations

journal contribution
posted on 2025-04-11, 00:20 authored by Tumadhir Fahim M ALSULAMITumadhir Fahim M ALSULAMI, Marcel JacksonMarcel Jackson

Despite the ubiquity of the rings ℀𝑛 as the finite, cyclic models of integer arithmetic, it is rather less well-known that cyclic models of (positive) arithmetic with exponentiation exist only for cycle length 1, 2, 6, 42, and 1806. We explore finite cyclic models of other arithmetical and combinatorial operations on positive arithmetic: fixed base exponentiation, factorial, and binomial coefficients. In each case we find whether infinitely or finitely many models are possible. The case of fixed base exponentiation is particularly interesting, where for base 𝑏 > 2 we find that the compatible cycle sizes form a multiplicative monoid of positive integers, with infinitely many irreducible elements, and curiously sparse prime factors.

History

Publication Date

2025-04-01

Journal

The American Mathematical Monthly

Volume

132

Issue

4

Pagination

316-332

Publisher

Taylor & Francis

ISSN

0002-9890

Rights Statement

Β© The Mathematical Association of America [2025] This is an Accepted Manuscript version of the following article, accepted for publication in The American Mathematical Monthly. Alsulami, T., & Jackson, M. (2025). Cyclic Arithmetic and Combinatorial Operations. The American Mathematical Monthly, 132(4), 316–332. https://doi.org/10.1080/00029890.2024.2439803. It is deposited under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.

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