New congruences modulo powers of 2 and 3 for 9-regular partitions

Abstract

Let b9(n) denote the number of 9-regular partitions of n. Recently, employing the theory of modular forms, Keith established several congruences modulo 2 and 3 for b9(n). He also presented four conjectures on b9(n) and two of them have been proved by Lin, and Xia and Yao. The remaining two conjectures are b9(32n+13)≡0(mod12) and b9(64n+13)≡0(mod24) for n⩾0. In this paper, employing 2-dissection formulas for certain quotients of theta functions, we prove that b9(32n+13)≡0(mod4) and b9(64n+13)≡0(mod8) for n⩾0. Combining these two congruences and the congruence b9(16n+13)≡0(mod3) proved by Keith, we confirm the remaining two conjectures of Keith. We also establish two infinite families of congruences modulo 9 for b9(n). For example, we prove that for all integers n⩾0 and k⩾1, b9(26kn+5×26k−1−13)≡0(mod9).