# Algorithms for the Multiplication Table Problem

## TL;DR

This paper explores algorithms for computing the number of distinct entries in the multiplication table, introducing a new subquadratic exact algorithm and Monte Carlo methods for approximation, with extensive computational results.

## Contribution

It presents a new subquadratic algorithm for exact computation of M(n) and two Monte Carlo algorithms for approximation, along with large-scale computational experiments.

## Key findings

- Exact computations for n up to 2^30
- Monte Carlo approximations for n up to 2^100,000,000
- Experimental results align with Ford's asymptotic estimate

## Abstract

Let $M(n)$ denote the number of distinct entries in the $n \times n$ multiplication table. The function $M(n)$ has been studied by Erd\H{o}s, Tenenbaum, Ford, and others, but the asymptotic behaviour of $M(n)$ as $n \to \infty$ is not known precisely. Thus, there is some interest in algorithms for computing $M(n)$ either exactly or approximately. We compare several algorithms for computing $M(n)$ exactly, and give a new algorithm that has a subquadratic running time. We also present two Monte Carlo algorithms for approximate computation of $M(n)$. We give the results of exact computations for values of $n$ up to $2^{30}$, and of Monte Carlo computations for $n$ up to $2^{100,000,000}$, and compare our experimental results with Ford's order-of-magnitude result.