Two Games on Arithmetic Functions: SALIQUANT and NONTOTIENT

We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in \cite{sum}. In each game, the set of positions is N (natural numbers). In saliquant, the options are to subtract a non-divisor. Here we obtain sev...

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Bibliographic Details
Published in:arXiv.org 2023-09
Main Authors: Ellis, Paul, Shi, Jason, Thanatipanonda, Thotsaporn Aek, Tu, Andrew
Format: Article
Language:English
Subjects:
Arithmetic
Games
Number theory
Online Access:Get full text
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Summary:We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in \cite{sum}. In each game, the set of positions is N (natural numbers). In saliquant, the options are to subtract a non-divisor. Here we obtain several nice number theoretic lemmas, a fundamental theorem, and two conjectures about the eventual density of Sprague-Grundy values. In nontotient, the only option is to subtract the number of relatively prime residues. Here are able to calculate certain Sprague-Grundy values, and start to understand an appropriate class function.
ISSN:2331-8422