Counting certain quadratic partitions of zero modulo a prime number

Consider an odd prime number p≡2(mod3)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/pZ{\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and nu...

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Autores principales:	Xiao Wang, Li Aihua
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	partition of a number dirichlet character sum diophantine equation solution number 11d45 11p83 11l10 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/5db4c9befe3a4d158b6b5ee1e919e6f9
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spelling	oai:doaj.org-article:5db4c9befe3a4d158b6b5ee1e919e6f92021-12-05T14:10:52ZCounting certain quadratic partitions of zero modulo a prime number2391-545510.1515/math-2021-0032https://doaj.org/article/5db4c9befe3a4d158b6b5ee1e919e6f92021-05-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0032https://doaj.org/toc/2391-5455Consider an odd prime number p≡2(mod3)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/pZ{\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/pZ{\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 pp. 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 x2+y2+z2=0{x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z/pZ{\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.Xiao WangLi AihuaDe Gruyterarticlepartition of a numberdirichlet character sumdiophantine equationsolution number11d4511p8311l10MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 198-211 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	partition of a number dirichlet character sum diophantine equation solution number 11d45 11p83 11l10 Mathematics QA1-939
spellingShingle	partition of a number dirichlet character sum diophantine equation solution number 11d45 11p83 11l10 Mathematics QA1-939 Xiao Wang Li Aihua Counting certain quadratic partitions of zero modulo a prime number
description	Consider an odd prime number p≡2(mod3)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/pZ{\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/pZ{\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 pp. 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 x2+y2+z2=0{x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z/pZ{\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.
format	article
author	Xiao Wang Li Aihua
author_facet	Xiao Wang Li Aihua
author_sort	Xiao Wang
title	Counting certain quadratic partitions of zero modulo a prime number
title_short	Counting certain quadratic partitions of zero modulo a prime number
title_full	Counting certain quadratic partitions of zero modulo a prime number
title_fullStr	Counting certain quadratic partitions of zero modulo a prime number
title_full_unstemmed	Counting certain quadratic partitions of zero modulo a prime number
title_sort	counting certain quadratic partitions of zero modulo a prime number
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/5db4c9befe3a4d158b6b5ee1e919e6f9
work_keys_str_mv	AT xiaowang countingcertainquadraticpartitionsofzeromoduloaprimenumber AT liaihua countingcertainquadraticpartitionsofzeromoduloaprimenumber
_version_	1718371645224124416

Counting certain quadratic partitions of zero modulo a prime number

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