Three Results on Making Change (An Exposition)

TL;DR

This paper presents a clear exposition of three theorems related to the number of ways to make change for n cents using a set of coin denominations, highlighting polynomial and asymptotic properties.

Contribution

It provides a unified, simplified explanation of three known theorems about the function CH(n), including polynomial and asymptotic behaviors, with a focus on their mathematical relationships.

Findings

CH(n) restricted to certain residue classes is a polynomial.

CH(n) exhibits polynomial behavior with exceptions in constant terms.

Asymptotically, CH(n) grows like n^{L-1}/((L-1)!a1a2...aL).

Abstract

Assume you an infinite supply of pennies, nickels, dimes, and quarters (or some other finite set of denominations which are relatively prime). Let CH(n) be the number of ways to make change of n cents. We present a simple unified exposition of three know theorems about CH(n). Let M be the LCM of a1,...,aL. Let M' be the LCM of the GCD of all pairs of ai's. (1) If 0\le r\le M-1 then CH(n) restricted to n \equiv r mod M is a poly, (2) If 0\le r\le M'-1 then CH(n) restricted to n\equiv r mod M' is a poly except for the constant term, (3) CH(n) is n^{L-1}/(L-1)!a1a2...aL + O(n^{L-2}). Part (3) is known as Schur's theorem.