Two Congruences Involving 4-cores

Abstract

The goal of this paper is to prove two new congruences involving 4–cores using elementary techniques; namely, if $a_4(n)$ denotes the number of 4–cores of $n$, then $a_4(9n+2)\equiv 0$ (mod 2) and $a_4(9n+8)\equiv 0$ (mod 4).