Related papers: On 3 and 9-regular cubic partitions

A cubic partition consists of partition pairs $(\lambda,\mu)$ such that $\vert\lambda\vert+\vert\mu\vert=n$ where $\mu$ involves only even integers but no restriction is placed on $\lambda$. This paper initiates the notion of generalized…

In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed $\alpha_2\in \mathbb{N}$ coprime to $q$, the congruence \[ x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q} \] has a solution of…

The primary focus of this paper is overpartitions, a type of partition that plays a significant role in $q$-series theory. In 2006, Treneer discovered an explicit infinite family of congruences of overpartitions modulo $5$. In our research,…

Let $a, b, c,$ and $n$ be integers, with $a$ nonzero and $n$ at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \]…

Let $A_{3}(n)$ (resp. ${{B}_{3}}(n)$) denote the number of partition pairs (resp. triples) of $n$ where each partition is 3-core. By applying Ramanujan's ${}_{1}\psi_{1}$ formula and Bailey's ${}_{6}\psi_{6}$ formula, we find the explicit…

Amdeberhan and Merca recently studied arithmetic properties of the sequence $a(n)$, the reciprocal of the crank parity function, which counts the number of integer partitions of weight $n$ whose even parts are monochromatic and whose odd…

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes.…

In this paper, we establish two new Ramanujan-type congruences for the overpartition function: $\overline{p}(11\times(8n+5))\equiv 0 \pmod{11}$ and $\overline{p}(13\times 2^6(8n+7))\equiv 0 \pmod{13}$. The proofs rely on the theory of…

In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for…

The starting point for this work is the family of functions $\overline{p}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by $\overline{p}_{-t}(n)$…

Let $p(n)$ be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers $N$ for which $p(N)$ is not congruent to $0\pmod{3}$. Radu proved this conjecture in 2010 using work of…

This paper confirms a conjecture of Amdeberhan and Moll that the power of 2 dividing the number of plane partitions in an n-cube is greater than the power of 2 dividing the number of totally symmetric plane partitions in the same cube when…

In this paper, we consider the set of partitions $pend(n)$ which enumerates the number of partitions of $n$ wherein the even parts are not allowed to be distinct. Using a result of Newman, we prove a few infinite families of congruences…

We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for…

Let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of weight $n$. Recently, Hirschhorn and Sellers, Yao, and Xia established a number of congruences modulo 2 and 5, 4 and 8, and 25 for $f(n)$, respectively. In this…

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q…

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

Let ${p}_{3,3}(n)$ denote the number of $2$-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$. In this paper, we shall establish some interesting Ramanujan-type congruences for…

Let $\{A'_n\}$ be the Ap\'ery numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be…