The existence of irrational most perfect magic squares

IF 0.5 4区数学 Q3 MATHEMATICS Journal of Combinatorial Designs Pub Date : 2022-11-09 DOI:10.1002/jcd.21865

Jingyuan Chen, Jinwei Wu, Dianhua Wu

{"title":"The existence of irrational most perfect magic squares","authors":"Jingyuan Chen, Jinwei Wu, Dianhua Wu","doi":"10.1002/jcd.21865","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>≡</mo>\n \n <mn>0</mn>\n <mspace></mspace>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>mod</mi>\n <mspace></mspace>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0001\" wiley:location=\"equation/jcd21865-math-0001.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x02261}</mo><mn>0</mn><mspace width=\"0.3em\"/><mrow><mo class=\"MathClass-open\">(</mo><mrow><mi>mod</mi><mspace width=\"0.3em\"/><mn>4</mn></mrow><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> be a positive integer, <math>\n <semantics>\n <mrow>\n <mi>M</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0002\" wiley:location=\"equation/jcd21865-math-0002.png\"><mrow><mrow><mi>M</mi><mo>=</mo><mrow><mo class=\"MathClass-open\">(</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> be a magic square, where <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n \n <mo>≤</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>≤</mo>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>−</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>0</mn>\n \n <mo>≤</mo>\n \n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n \n <mo>≤</mo>\n \n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0003\" wiley:location=\"equation/jcd21865-math-0003.png\"><mrow><mrow><mn>0</mn><mo>\\unicode{x02264}</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>\\unicode{x02264}</mo><msup><mi>n</mi><mn>2</mn></msup><mo>\\unicode{x02212}</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>\\unicode{x02264}</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>\\unicode{x02264}</mo><mi>n</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>. <math>\n <semantics>\n <mrow>\n <mi>M</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0004\" wiley:location=\"equation/jcd21865-math-0004.png\"><mrow><mrow><mi>M</mi></mrow></mrow></math></annotation>\n </semantics></math> is called most perfect magic square (MPMS<math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mi>n</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0005\" wiley:location=\"equation/jcd21865-math-0005.png\"><mrow><mrow><mrow><mo class=\"MathClass-open\">(</mo><mi>n</mi><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> for short) if <math>\n <semantics>\n <mrow>\n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>+</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>+</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mn>2</mn>\n </mfrac>\n \n <mo>,</mo>\n \n <mi>j</mi>\n \n <mo>+</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mn>2</mn>\n </mfrac>\n </mrow>\n </msub>\n \n <mo>=</mo>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0006\" wiley:location=\"equation/jcd21865-math-0006.png\"><mrow><mrow><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>\\unicode{x0002B}</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>\\unicode{x0002B}</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>,</mo><mi>j</mi><mo>\\unicode{x0002B}</mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></msub><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mo>\\unicode{x02212}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>+</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>+</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n <mspace></mspace>\n \n <mo>+</mo>\n \n <msub>\n <mi>m</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mi>j</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>=</mo>\n \n <mn>2</mn>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0007\" wiley:location=\"equation/jcd21865-math-0007.png\"><mrow><mrow><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>\\unicode{x0002B}</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub><mo>\\unicode{x0002B}</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>\\unicode{x0002B}</mo><mn>1</mn><mo>,</mo><mi>j</mi></mrow></msub><mspace width=\"0.25em\"/><mo>\\unicode{x0002B}</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>\\unicode{x0002B}</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mrow><mo class=\"MathClass-open\">(</mo><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>. Let <math>\n <semantics>\n <mrow>\n <mi>M</mi>\n \n <mo>=</mo>\n \n <mi>n</mi>\n \n <mi>A</mi>\n \n <mo>+</mo>\n \n <mi>B</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0008\" wiley:location=\"equation/jcd21865-math-0008.png\"><mrow><mrow><mi>M</mi><mo>=</mo><mi>n</mi><mi>A</mi><mo>\\unicode{x0002B}</mo><mi>B</mi></mrow></mrow></math></annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>a</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>B</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>b</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mn>0</mn>\n \n <mo>≤</mo>\n \n <msub>\n <mi>a</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>b</mi>\n \n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n </msub>\n \n <mo>≤</mo>\n \n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0009\" wiley:location=\"equation/jcd21865-math-0009.png\"><mrow><mrow><mi>A</mi><mo>=</mo><mrow><mo class=\"MathClass-open\">(</mo><msub><mi>a</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo class=\"MathClass-close\">)</mo></mrow><mo>,</mo><mi>B</mi><mo>=</mo><mrow><mo class=\"MathClass-open\">(</mo><msub><mi>b</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo class=\"MathClass-close\">)</mo></mrow><mo>,</mo><mn>0</mn><mo>\\unicode{x02264}</mo><msub><mi>a</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>b</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>\\unicode{x02264}</mo><mi>n</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>. <math>\n <semantics>\n <mrow>\n <mi>M</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0010\" wiley:location=\"equation/jcd21865-math-0010.png\"><mrow><mrow><mi>M</mi></mrow></mrow></math></annotation>\n </semantics></math> is called rational if both <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0011\" wiley:location=\"equation/jcd21865-math-0011.png\"><mrow><mrow><mi>A</mi></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>B</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0012\" wiley:location=\"equation/jcd21865-math-0012.png\"><mrow><mrow><mi>B</mi></mrow></mrow></math></annotation>\n </semantics></math> possess the property that the sums of the numbers in every row and every column are the same; otherwise, <math>\n <semantics>\n <mrow>\n <mi>M</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0013\" wiley:location=\"equation/jcd21865-math-0013.png\"><mrow><mrow><mi>M</mi></mrow></mrow></math></annotation>\n </semantics></math> is said to be irrational. It was shown that there exists an MPMS<math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mi>n</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0014\" wiley:location=\"equation/jcd21865-math-0014.png\"><mrow><mrow><mrow><mo class=\"MathClass-open\">(</mo><mi>n</mi><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> if and only if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>≡</mo>\n \n <mn>0</mn>\n <mspace></mspace>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>mod</mi>\n <mspace></mspace>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0015\" wiley:location=\"equation/jcd21865-math-0015.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x02261}</mo><mn>0</mn><mspace width=\"0.3em\"/><mrow><mo class=\"MathClass-open\">(</mo><mrow><mi>mod</mi><mspace width=\"0.3em\"/><mn>4</mn></mrow><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>≥</mo>\n \n <mn>4</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0016\" wiley:location=\"equation/jcd21865-math-0016.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x02265}</mo><mn>4</mn></mrow></mrow></math></annotation>\n </semantics></math>. In this paper, it is proved that there exists an irrational MPMS<math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mi>n</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0017\" wiley:location=\"equation/jcd21865-math-0017.png\"><mrow><mrow><mrow><mo class=\"MathClass-open\">(</mo><mi>n</mi><mo class=\"MathClass-close\">)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> if and only if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>=</mo>\n \n <msup>\n <mn>2</mn>\n \n <mi>t</mi>\n </msup>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>t</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>≡</mo>\n \n <mn>1</mn>\n <mspace></mspace>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>mod</mi>\n <mspace></mspace>\n \n <mn>2</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0018\" wiley:location=\"equation/jcd21865-math-0018.png\"><mrow><mrow><mi>n</mi><mo>=</mo><msup><mn>2</mn><mi>t</mi></msup><mi>k</mi><mo>,</mo><mi>t</mi><mo>\\unicode{x02265}</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo>\\unicode{x02261}</mo><mn>1</mn><mspace width=\"0.3em\"/><mrow><mo class=\"MathClass-open\">(</mo><mrow><mi>mod</mi><mspace width=\"0.3em\"/><mn>2</mn></mrow><mo class=\"MathClass-close\">)</mo></mrow><mo>,</mo></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>></mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0019\" wiley:location=\"equation/jcd21865-math-0019.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x0003E}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 1","pages":"23-40"},"PeriodicalIF":0.5000,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21865","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 1

Abstract

Let $n \equiv 0 (\mod 4)$ be a positive integer, $M = (m_{i, j})$ be a magic square, where $0 \leq m_{i, j} \leq n^{2} - 1, 0 \leq i, j \leq n - 1$ . $M$ is called most perfect magic square (MPMS $(n)$ for short) if $m_{i, j} + m_{i + \frac{n}{2}, j + \frac{n}{2}} = n^{2} - 1$ , and $m_{i, j} + m_{i, j + 1} + m_{i + 1, j} + m_{i + 1, j + 1} = 2 (n^{2} - 1)$ . Let $M = n A + B$ , where $A = (a_{i, j}), B = (b_{i, j}), 0 \leq a_{i, j}, b_{i, j} \leq n - 1$ . $M$ is called rational if both $A$ and $B$ possess the property that the sums of the numbers in every row and every column are the same; otherwise, $M$ is said to be irrational. It was shown that there exists an MPMS $(n)$ if and only if $n \equiv 0 (\mod 4)$ and $n \geq 4$ . In this paper, it is proved that there exists an irrational MPMS $(n)$ if and only if $n = 2^{t} k, t \geq 2, k \equiv 1 (\mod 2),$ and $k > 1$ .

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

非理性最完美幻方的存在性

PNG“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；msub&gt；&lt；mi&gt；i&lt；mi&gt；&lt；mo&gt；&lt；j&lt；mi&gt；&lt；&lt；mrow&gt；&lt；msub&gt；&lt；mo&gt；unicode{x0002b}&lt；&lt；mo&gt；&lt；msub&gt；&lt；&lt；mi&gt；M&lt；mi&gt；&lt；mi&gt；i&lt；mi&gt；&lt；mo&gt；&lt；mi&gt；j&lt；mi&gt；&lt；mo&gt；unicode{x0002b}&lt；mo&gt；&lt；mn&gt；1&lt；mn&gt；&lt；mrow&gt；&lt；msub&lt；mo&gt；unicode{x0002b}&lt；/MO&gt；&lt；MSUB&gt；&lt；MI&gt；m&lt；/MI&gt；&lt；mrow&gt；&lt；MI&gt；I&lt；/MI&gt；&lt；MO&gt；\unicode{x0002b}&lt；/MO&gt；&lt；Mn&gt；1&lt；/MN&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；MI&gt；J&lt；/MI&gt；&lt；/mrow&gt；&lt；/MSUB&gt；&lt；mspace width=“0.25em”/&gt；lt；MO&gt；\unicode{x0002b}&lt；/MO&gt；&lt；MSUB&gt；&lt；MI&gt；m&lt；/MI&gt；&lt；mrow&gt；&lt；MI&gt；I&lt；/MI&gt；&lt；MO&gt；\unicode{x0002b}&lt；/MO&gt；&lt；Mn&gt；1&lt；/MN&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；MI&gt；J&lt；/MI&gt；&lt；MO&gt；\unicode{x0002b}&lt；/MO&gt；&lt；Mn&gt；1&lt；/MN&gt；&lt；/mrow&gt；&lt；/MSUB&gt；&lt；mo&gt；=&lt；/MO&gt；&lt；Mn&gt；2&lt；/MN&gt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；mo&gt；&lt；mrow&gt；&lt；msup&gt；&lt；mi&gt；n&lt；mi&gt；&lt；mn&gt；2&lt；mn&gt；&lt；msup&gt；&lt；mo&gt；unicode{x02212}&lt；mo&gt；&lt；mn&gt；1&lt；mn&gt；&lt；mrow&gt；&lt；mo class=“mathclass close”&gt；）&lt；/MO&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；。let m=n a+b&lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0008“wiley:location=”equation/jcd21865-math-0008.png“&gt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；m&lt；mi&gt；&lt；mo&gt；&lt；mi&gt；n&lt；mi&gt；a&lt；mi&gt；&lt；mo&gt；\unics ode{x0002b}&lt；mo&gt；mi&gt；b&lt；mi&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；math&gt；，其中a=（A i，j），B=（B i，j），0≤ai，j，b i，j≤n−1&lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0009“wiley:location=”equation/jcd21865-math-0009。 PNG“&gt；&lt；mrow&gt；&lt；mi&gt；a&lt；mi&gt；&lt；mo&gt；=&lt；mo&gt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；mo&gt；&lt；msub&gt；&lt；mi&gt；a&lt；mi&gt；&lt；mrow&gt；&lt；i&lt；mi&gt；&lt；mo&gt；&lt；j&lt；/mi&gt；&lt；mrow&gt；&lt；msub&gt；&lt；mo class=“mathclass close”&gt；）&lt；mo&gt；&lt；mrow&gt；&lt；mo&gt；&lt；&lt；mi&gt；b&lt；mi&gt；&lt；mo&gt；&lt；mo&gt；&lt；&lt；mo&gt；&lt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；mo&gt；&lt；mi&gt；b&lt；/mi&gt；&lt；mrow&gt；&lt；mi&gt；i&lt；/mi&gt；&lt；mo&gt；&lt；mi&gt；j&lt；/mi&gt；&lt；mrow&gt；&lt；mo class=“mathclass close”）&lt；/MO&gt；&lt；/mrow&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；Mn&gt；0&lt；/MN&gt；&lt；MO&gt；\unicode{x02264}&lt；/MO&gt；&lt；MSUB&gt；&lt；MI&gt；a&lt；/MI&gt；&lt；mrow&gt；&lt；MI&gt；I&lt；/MI&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；MI&gt；J&lt；/MI&gt；&lt；/mrow&gt；&lt；/MSUB&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；MSUB&gt；&lt；MI&gt；b&lt；/MI&gt；&lt；mrow&gt；&lt；MI&gt；I&lt；/MI&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；MI&gt；J&lt；/MI&gt；&lt；/mrow&gt；&lt；/MSUB&gt；&lt；MO&gt；\unicode{x02264}&lt；/MO&gt；&lt；MI&gt；n&lt；/MI&gt；&lt；MO&gt；\unicode{x02212}&lt；/MO&gt；&lt；Mn&gt；1&lt；/MN&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；。M&lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0010“wiley:location=”equation/jcd21865-math-0010.png“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；m&lt；mi&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；如果两个都有&lt；math xmlns=”http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0011“wiley:location=”equation/jcd21865-math-0011.png“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；a&lt；mi&gt；&lt；mrow&gt；&lt；和b&lt；math xmlns=”http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0012“wiley:location=”equation/jcd21865-math-0012.png“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；b&lt；mi&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；math&gt；拥有每行和每列中数字之和相同的属性；否则e、m&lt；math xmlns=”http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0013“wiley:location=”equation/jcd21865-math-0013.png“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；m&lt；mi&gt；&lt；&lt；mrow&gt；&lt；mrow&gt；&lt；/math&gt；据说是非理性的。这表明存在MPM（n）&lt；math xmlns=”http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0014“wiley:location=“equation/jcd21865-math-0014.png”&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；/mo&gt；&lt；mi&gt；n&lt；/mi&gt；&lt；mo class=“mathclass close”&gt；）&lt；/MO&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；如果且仅当n≥0（mod 4）lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0015“wiley:location=”equation/jcd21865-math-0015.png“&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；n&lt；mi&gt；&lt；mo&gt；&lt；unicode{x02261}&lt；mo&gt；&lt；mn&gt；0&lt；mn&gt；&lt；mspace width=“0.3em”/&gt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；&lt；/mo&gt；&lt；mrow&gt；&lt；mi&gt；mod&lt；/mi&gt；&lt；mspace width=“0.3em”/&gt；&lt；mn&gt；4&lt；/mn&gt；&lt；mrow&gt；&lt；mo class=“mathclass close”&gt；）&lt；/MO&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；且n≥4&lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0016”wiley:location=“equation/jcd21865-math-0016.png”&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；n&lt；mi&gt；&lt；mo&gt；&lt；unicode{x02265}&lt；mo&gt；&lt；mn&gt；4&lt；mn&gt；&lt；&lt；mrow&gt；&lt；/mrow&gt；&“数学”。在本文中，证明存在非理性MPM（n）；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0017“wiley:location=“equation/jcd21865-math-0017.png”&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；mo&gt；&lt；mi&gt；n&lt；/mi&gt；&lt；mo class=“mathclass close”&gt；）&lt；/MO&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；如果且仅当n=2tk，t≥2时，k≤1（mod 2），＜；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0018“wiley:location=”equation/jcd21865-math-0018.png“&gt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；n&lt；mi&gt；&lt；mo&gt；&lt；msup&gt；&lt；mn&gt；2&lt；mn&gt；&lt；mi&gt；t&lt；mi&gt；&lt；msup&lt；mi&gt；k&lt；mi&gt；mo&gt；&lt；mi&gt；t&lt；mi&gt；&lt；mo&gt；unicode{x02265}&lt；mo&gt；&lt；mn&gt；2&lt；mn&gt；&lt；mo&gt；&lt；k&lt；mi&gt；&lt；mo&gt；unicode{x02261}&lt；/MO&gt；&lt；Mn&gt；1&lt；/MN&gt；&lt；mspace width=“0.3em”/&gt；lt；mrow&gt；&lt；mo class=“mathclass open”&gt；（&lt；mo&gt；&lt；mrow&gt；&lt；mi&gt；mod&lt；/mi&gt；&lt；mspace width=“0.3em”/&gt；&lt；mn&gt；2&lt；mn&gt；&lt；mrow&gt；&lt；mo class=“mathclass close”&gt；）&lt；/MO&gt；&lt；/mrow&gt；&lt；mo&gt；，&lt；/MO&gt；&lt；/mrow&gt；&lt；/mrow&gt；&lt；/数学&gt；和k&gt；1&lt；数学xmlns=“http://www.w3.org/1998/Math/MathML“altimg=”urn:x-wiley:10638539:media:jcd21865:jcd21865-math-0019“wiley:location=“equation/jcd21865-math-0019.png”&gt；&lt；mrow&gt；&lt；mrow&gt；&lt；mi&gt；k&lt；mi&gt；&lt；mo&gt；&lt；unicode{x0003e}&lt；mo&gt；&lt；mn&gt；1&lt；mn&gt；&lt；&lt；mrow&gt；&lt；/mrow&gt；&“数学”。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.

期刊最新文献

Issue Information Extensions of Steiner Triple Systems On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays Avoiding Secants of Given Size in Finite Projective Planes Issue Information