Magic squares of squares over a finite field

IF 0.3 4区 数学 Q4 MATHEMATICS
S. Hengeveld, Giancarlo Labruna, Aihua Li
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Hengeveld, Giancarlo Labruna, Aihua Li","doi":"10.1090/conm/773/15536","DOIUrl":null,"url":null,"abstract":"<p>A magic square <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over an integral domain <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3 times 3\"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">3\\times 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix with entries from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the elements from each row, column, and diagonal add to the same sum. If all the entries in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are perfect squares in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we call <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a magic square of squares over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the integers which has all the nine entries distinct?” We approach to answering a similar question when <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite field. We claim that for any odd prime <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a magic square over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Subscript p\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathbb Z_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that, over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Subscript p\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathbb Z_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, magic squares of squares with nine distinct elements exist. In addition, if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p identical-to 1 left-parenthesis mod 120 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mn>1</mml:mn> <mml:mspace width=\"0.667em\" /> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=\"0.333em\" /> <mml:mn>120</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p\\equiv 1\\pmod {120}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exist magic squares of squares over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Subscript p\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathbb Z_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.</p>","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"39 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Commutative Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/conm/773/15536","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

有限域上正方形的幻方
在积分域上的幻方是一个3 × 3 × 3 × 3的矩阵,它的元素来自于D D使得每一行,每一列,每一条对角线的元素相加到相同的和。如果M M中的所有项都是D D中的完全平方项,我们称M M为D D上的平方的幻方。1984年,Martin LaBar提出了一个开放的问题:“在整数环Z \mathbb {Z}上是否存在一个魔方的平方,其中所有的九个元素都是不同的?”当pdd是有限域时,我们试图回答一个类似的问题。我们声明对于任何奇数素数p p,一个幻方除以zp \mathbb Z_p只能包含奇数个不同的元素。对应于LaBar的问题,我们证明了存在无穷多个素数p p,使得在zp \mathbb Z_p上,有九个不同元素的平方的幻方存在。另外,如果p≡1 (mod 120) p\equiv 1\pmod{120},则存在Z p\ mathbb Z_p上的平方的幻方,它们分别有3、5、7或9个不同的项。我们用由孪生素数导出的连续二次残的三元组构造幻方的幻方。
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来源期刊
CiteScore
0.80
自引率
16.70%
发文量
28
审稿时长
>12 weeks
期刊介绍: Journal of Commutative Algebra publishes significant results in the area of commutative algebra and closely related fields including algebraic number theory, algebraic geometry, representation theory, semigroups and monoids. The journal also publishes substantial expository/survey papers as well as conference proceedings. Any person interested in editing such a proceeding should contact one of the managing editors.
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