An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions

Abstract

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c{\varphi }_k(n)$ where $k\ge 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\ge 0,$ $c{\varphi }_2(5n+3)\equiv 0(\mathrm{mod}5).$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. In 2011, Baruah and Sarmah proved a number of congruence properties for $c{\varphi }_4$, all with moduli which are powers of 4. In this brief note, we add to the collection of congruences for $c{\varphi }_4$ by proving this function satisfies an unexpected result modulo 5. The proof is elementary, relying on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan.

References in corpus (1)

• Congruences modulo powers of 5 for three-colored Frobenius partitions

Cited by in corpus (4)

• Formulas for Jacobi forms and generalized Frobenius partitions
• Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors
• Congruences Modulo Powers of 3 for 3- and 9-Colored Generalized Frobenius Partitions
• Modular Forms and $k$-colored Generalized Frobenius Partitions