Congruences Modulo Powers of 3 for 3- and 9-Colored Generalized Frobenius Partitions

2018/01/24 by Wang, Liuquan
#05A17 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1801.07949

Abstract

Let cϕ_k(n) be the number of k-colored generalized Frobenius partitions of n. We establish some infinite families of congruences for cϕ₃(n) and cϕ₉(n) modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for k≥ 3 and n≥ 0, we prove that cϕ₃(3^2kn+\frac7⋅ 3^2k+18) ≡ 0 \pmod3^4k+5. We give two different proofs to the congruences satisfied by cϕ₉(n). One of the proofs uses an relation between cϕ₉(n) and cϕ₃(n) due to Kolitsch, for which we provide a new proof in this paper.

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