Let b(n) denote the number of cubic partition pairs of n. We affirm a conjecture of Lin by proving that$$\begin{aligned} b(49n+37)\equiv 0 \pmod {49} \end{aligned}$$for all \(n\ge 0\). We also prove two congruences modulo 256 satisfied by \(\overline{b}(n)\), the number of overcubic partition pairs of n. Let \(\overline{a}(n)\) denote the number of overcubic partition of n. For any positive integer k, we show that \(\overline{b}(n)\) and \(\overline{a}(n)\) are divisible by \(2^k\) for almost all n. We use arithmetic properties of modular forms to prove our results.

中文翻译：

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三次和三次分区对的算术性质

让b（Ñ）表示的立方分区对数Ñ。我们通过证明对所有\（n \ ge 0 \）的$$ \ begin {aligned} b（49n + 37）\ equiv 0 \ pmod {49} \ end {aligned} $$来肯定Lin的猜想。我们还证明了2个同余模256满足\（\划线{B}（N）\）的overcubic分区对数ñ。令\（\ overline {a}（n）\）表示n的超三次分区数。对于任何正整数ķ，我们表明，\（\划线{B}（N）\）和\（\划线{A}（N）\）是整除\（2 ^ķ\）对于几乎所有n。我们使用模块化形式的算术特性来证明我们的结果。