{"title":"Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)","authors":"Yasutsugu Fujita , Maohua Le","doi":"10.1016/j.jcta.2024.106004","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span>, <span><math><mi>P</mi></math></span> be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma <span><span>[14]</span></span> conjectured that the equation <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>p</mi><mo>∈</mo><mi>P</mi><mo>,</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> has only one solution <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>49</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is a solution of <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> with <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, then there exist an odd positive integer <em>g</em> and a positive integer <em>t</em> which make <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi></mrow></msup><mo>=</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mo>(</mo><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>/</mo><mo>(</mo><mi>α</mi><mo>−</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> for any integer <em>r</em>, <span><math><mi>α</mi><mo>=</mo><mn>2</mn><mi>g</mi><mo>+</mo><msqrt><mrow><mn>4</mn><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></math></span> and <span><math><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>=</mo><mn>2</mn><mi>g</mi><mo>−</mo><msqrt><mrow><mn>4</mn><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></math></span>. Then, we obtain certain properties of the positive integers <em>t</em> and <em>g</em>. Finally, we comprehensively apply some classical results from transcendental number theory and Diophantine equations to prove that, for any fixed odd prime <em>p</em>, all solutions <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> of <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> with <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span> satisfy <span><math><mi>x</mi><mo><</mo><mi>C</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>, where <span><math><mi>C</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span> is an effectively computable constant depending only on <em>p</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106004"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524001432","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let , be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma [14] conjectured that the equation , has only one solution . This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if is a solution of with , then there exist an odd positive integer g and a positive integer t which make and , where for any integer r, and . Then, we obtain certain properties of the positive integers t and g. Finally, we comprehensively apply some classical results from transcendental number theory and Diophantine equations to prove that, for any fixed odd prime p, all solutions of with satisfy , where is an effectively computable constant depending only on p.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.