Game values of arithmetic functions

TL;DR

This paper explores the intersection of number theory and combinatorial game theory by analyzing two-player games based on arithmetic functions like division, divisors, and coprimality, revealing new insights into their strategic properties.

Contribution

It introduces a novel framework connecting arithmetic functions with game values, providing a systematic study of their associated combinatorial games.

Findings

Characterization of game values for division and divisor-based games

Identification of Sprague-Grundy functions for various arithmetic-based games

Insights into strategic complexity of number-theoretic games

Abstract

Arithmetic functions in Number Theory meet the Sprague-Grundy function from Combinatorial Game Theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and relatively prime numbers, and their negations.