Periodicity of A Partition Function Related to Making Change Modulo Prime Powers

Abstract

Let \(p_e(n)\) be the number of ways to make change for \(n\) cents using pennies, nickels, dimes, and quarters. By manipulating the generating function for \(p_e(n)\), we prove that the sequence \(\{p_e(n) \pmod{\ell^j}\}\) is periodic for every prime power \(\ell\).