论文标题
3型分区编号和二元方程的均等
Parity of 3-regular partition numbers and Diophantine equations
论文作者
论文摘要
令$ b_3（n）$为$ 3 $ n $的$ 3 $定型分区的数量。最近，W。J。Keith和F. Zanello发现了Ramanujan类型的无限家族，$ 2 $ 2 $ $ 2 $，$ b_3（2n）$，涉及每个Prime $ P $，带有$ P \ equiv 13，17，19，19，23，23，23 \ pmod {24} $ $ b_3（2n）$涉及每个prime $ p \ geqslant 5 $。在本文中，我们介绍了新的无限Ramanujan类型一致性Modulo $ 2 $ $ b_3（2n）$。他们自然地补充了Keith-Zanello和Yao的结果，并涉及$ \ Mathcal p = \ {p \ {p \ text {prips {priper}：\ exists \，j \ in \ in \ {1,4,8 \}，\，\，\，\，x，x，x，y \ in \ in \ mathb z，\ n \ nathbb z，\，\ y，x，y} x^2+216y^2 = jp \} $，其dirichlet密度为$ 1/6 $。作为证明的关键成分，我们证明了$ x^2+216y^2 = pm $，$ p \ in \ Mathcal p $，$ p \ nmid m $和$ pm \ equiv 1 \ equiv 1 \ pmod {24} $的原始解决方案数量，$ 8 $。在这里，困难源于以下事实：$ 216 $不是Idoneal。我们还提供了一个猜想的精确公式，用于该二磷酸方程的溶液数量。在本文的第二部分中，我们研究了欧拉型身份的逆转。这些是由第二作者最近的工作激发的，涉及Schur身份的逆转，该身份涉及$ 3 $的定型分区，并由其长度的奇偶校验加权。
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime } : \exists \, j\in \{1,4,8\},\, x, y \in \mathbb Z,\, \gcd(x,y)=1 \text { with } x^2+216y^2=jp\}$ whose Dirichlet density is $1/6$. As a key ingredient in our proof we show that of the number of primitive solutions for $x^2+216y^2=pm$, $p \in \mathcal P$, $p\nmid m$ and $pm\equiv 1\pmod{24}$, is divisible by $8$. Here, the difficulty arises from the fact that $216$ is not idoneal. We also give a conjectural exact formula for the number of solutions for this Diophantine equation. In the second part of the article, we study reversals of Euler-type identities. These are motivated by recent work of the second author on a reversal of Schur's identity which involves $3$-regular partitions weighted by the parity of their length.
