Congruences for 9-regular partitions modulo 3

Abstract

In view of the modular equation of fifth order, we give a simple proof of Keith's conjecture which is some infinite families of congruences modulo 3 for the 9-regular partition function. Meanwhile, we derive some new congruences modulo 3 for the 9-regular partition function.

Congruences for (3,11)-regular bipartitions modulo 11

“…Recently, Xia and Yao [31] established several infinite families of congruences modulo 2 for b 9 (n). On the other hand, the congruences modulo 3 for b 9 (n) have been studied by several authors, see [11,16,23,32,33].…”

Some New Congruences for l-Regular Partitions Modulo 13, 17, and 23

“…Furcy and Penniston [FuPe12] obtained families of congruences modulo 3 for other values of l which are congruent to 1 modulo 3. Xia and Yao [XiYa14a] found some infinite families of congruences for b 9 (n) modulo 2, and Cui and Gu [CuiGu15] derived congruences for b 9 (n) modulo 3. Xia [Xia15] established infinite families of congruences for b l (n) modulo l where l ∈ {13, 17, 19}, for example, for any n ≥ 0, k ≥ 0, In this paper, we establish infinite families of Ramanujan-type congruences for b l (n) modulo l, where l ∈ {17, 23}, and for b 65 (n) modulo 13.…”

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

“…As for the arithmetic properties of b (n) modulo 3, Cui and Gu [5] and Keith [9] and Xia and Yao [17] studied respectively the congruences for b 9 (n) modulo 3. Lin and Wang [11] showed that 9-regular partitions and 3-cores satisfy the same congruences modulo 3 and further generalized Keith's conjecture and derived a stronger result.…”