Arithmetic Upper and Lower Bounds for the Denumerant Function

TL;DR

This paper derives new tight upper and lower bounds for Sylvester's denumerant function using elementary arithmetic, enabling precise understanding of its asymptotic behavior.

Contribution

It introduces novel bounds for the denumerant function based on elementary methods, improving previous estimates and clarifying its asymptotic properties.

Findings

Bounds are tight enough for asymptotic analysis

Elementary arithmetic methods are effective for bounding the denumerant

Provides insights into the growth rate of the denumerant function

Abstract

We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denumerant function. These bounds are tight enough to get the asymptotic behavior of the denumerant.