Counting certain quadratic partitions of zero modulo a prime number

Abstract

Abstract 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.