非理性最完美幻方的存在性

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":"非理性最完美幻方的存在性","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":"{\"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}","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

摘要

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”。

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The existence of irrational most perfect magic squares

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 $k > 1$ .

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