论文标题
无限家族的基本证明是Merca的立方分区
Elementary Proofs of Infinite Families of Congruences for Merca's Cubic Partitions
论文作者
论文摘要
最近，使用模块化表单和Smoot的{\ tt Mathematica}实现了Radu的算法证明分区的一致性，Merca证明了以下两个一致性：对于所有$ n \ geq 0，$ n \ geq 0，$ \ begin {align*} \ pmod {3}。 \ end {align*}这里$ a（n）$与计算{\ it cubic分区}数量的函数密切相关，其中允许以两种不同颜色出现均匀零件的分区。实际上，$ a（n）$的定义是在偶数零件中的立方分区数量与$ n $的立方分区的数量之间的差异。 在此简短说明中，我们通过经典生成功能操作提供了这两个一致性的基本证明。然后，我们证明了两个无限的Ramanujan的无限家庭 - 像$ a（n）$满足的类似一致性Modulo 3，其中Merca的最初两个一致性是每个家庭的最初成员。
Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\ A(27n+26) & \equiv 0 \pmod{3}. \end{align*} Here $A(n)$ is closely related to the function which counts the number of {\it cubic partitions}, partitions wherein the even parts are allowed to appear in two different colors. Indeed, $A(n)$ is defined as the difference between the number of cubic partitions of $n$ into an even numbers of parts and the number of cubic partitions of $n$ into an odd numbers of parts. In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non--nested Ramanujan--like congruences modulo 3 satisfied by $A(n)$ wherein Merca's original two congruences serve as the initial members of each family.
