Date of Award

5-2024

Degree Type

Thesis

Degree Name

Honors Thesis

Department

Mathematics

First Advisor

Jonathan Webster

Second Advisor

Jon Sorenson

Abstract

In 1955, Paul Erdős initiated the study of a function that counts the number of distinct integers in an (n × n) multiplication table. That is, he studied M(n) = |{i · j, 1 ≤ i, j ≤ n}|. Much research has been done in regards to both asymptotic and exact approximations of M(n) for increasingly large values of n. Recently, Brent et. al. investigated the algorithmic cost in computing this function. Instead of computing M(n) directly, their approach was to compute it incrementally. That is, given M(n−1), they could quickly compute M(n) using another function δ(n) to count the number of distinct values in the newly added column of a table. We improve on their incremental result by providing a faster algorithm for a large subset of their cases. This is based on an understanding of smaller rectangular shapes in a multiplication table and was also studied, from a purely theoretical point of view, by Koukoulopoulos. We conclude by offering new fast evaluation results for δ(n) and showing a possible recursive method for computing M(n).

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