Dismal Arithmetic

TL;DR

This paper explores the properties of dismal arithmetic, a simplified number system where addition and multiplication are defined by taking the maximum and minimum digits respectively, and investigates its number theory analogues.

Contribution

It introduces and analyzes fundamental number theory concepts within the framework of dismal arithmetic, providing new insights into this simplified mathematical system.

Findings

Identification of prime analogues in dismal arithmetic

Formulas for the number of divisors and sum of divisors

Analysis of the partition function in dismal arithmetic

Abstract

Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number theory in this world, including analogues of the primes, number of divisors, sum of divisors, and the partition function.