Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

中文翻译：

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计算零模的质数为零的某些二次分区

考虑奇数质数p≡2（mod 3）p \ equiv 2 \ hspace {0.3em} \ left（\ mathrm {mod} \ hspace {0.3em} 3）。在本文中，Z / p Z {\ mathbb {Z}} \ hspace {-0.1em} \ text {/} \ hspace {-0.1em} p {\ mathbb {Z }}是结合基本组合和数论计算得出的。重点是Z / p Z中的0的三部分分区{\ mathbb {Z}} \ hspace {-0.1em} \ text {/} \ hspace {-0.1em} p {\ mathbb {Z}}从非零二次残基mod pp的集合中选择所有三个部分。这样的分区分为两种类型。具有完全相同的三个部分中的两个部分的分类为I型。类型II分区是所有三个部分都不同的分区。给出了每种类型的分区数。计算此类分区的问题与计算Diophantine方程x 2 + y 2 + z 2 = 0 {x} ^ {2} + {y} ^ {2} +的非平凡解的数量非常相关环Z / p Z中的{z} ^ {2} = 0 {\ mathbb {Z}} \ hspace {-0.1em} \ text {/} \ hspace {-0.1em} p {\ mathbb {Z}} 。相应地，该方程的解也被分类为I型或II型。我们给出与每种类型相对应的方程的解数。