{"title":"关于三元纯指数丢番图方程Ax + By = Cz与A + B = C2的注释","authors":"Elif Kizildere, M. Le, G. Soydan","doi":"10.1556/012.2020.57.2.1457","DOIUrl":null,"url":null,"abstract":"<jats:p>Let <jats:italic>l,m,r</jats:italic> be fixed positive integers such that 2<jats:inline-formula />| <jats:italic>l</jats:italic>, 3<jats:inline-formula /><jats:italic>lm</jats:italic>, <jats:italic>l > r</jats:italic> and 3 | <jats:italic>r</jats:italic>. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{<jats:italic>rlm</jats:italic><jats:sup>2</jats:sup> − 1<jats:italic>,</jats:italic>(<jats:italic>l</jats:italic> − <jats:italic>r</jats:italic>)<jats:italic>lm</jats:italic><jats:sup>2</jats:sup> + 1} <jats:italic>></jats:italic> 30, then the equation (<jats:italic>rlm</jats:italic><jats:sup>2</jats:sup> − 1)<jats:sup><jats:italic>x</jats:italic></jats:sup> + ((<jats:italic>l</jats:italic> − <jats:italic>r</jats:italic>)<jats:italic>lm</jats:italic><jats:sup>2</jats:sup> + 1)<jats:sup><jats:italic>y</jats:italic></jats:sup> = (<jats:italic>lm</jats:italic>)<jats:sup><jats:italic>z</jats:italic></jats:sup> has only the positive integer solution (<jats:italic>x,y,z</jats:italic>) = (1<jats:italic>,</jats:italic>1<jats:italic>,</jats:italic>2).</jats:p>","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"20 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"A note on the ternary purely exponential diophantine equation Ax + By = Cz with A + B = C2\",\"authors\":\"Elif Kizildere, M. Le, G. Soydan\",\"doi\":\"10.1556/012.2020.57.2.1457\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>Let <jats:italic>l,m,r</jats:italic> be fixed positive integers such that 2<jats:inline-formula />| <jats:italic>l</jats:italic>, 3<jats:inline-formula /><jats:italic>lm</jats:italic>, <jats:italic>l > r</jats:italic> and 3 | <jats:italic>r</jats:italic>. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{<jats:italic>rlm</jats:italic><jats:sup>2</jats:sup> − 1<jats:italic>,</jats:italic>(<jats:italic>l</jats:italic> − <jats:italic>r</jats:italic>)<jats:italic>lm</jats:italic><jats:sup>2</jats:sup> + 1} <jats:italic>></jats:italic> 30, then the equation (<jats:italic>rlm</jats:italic><jats:sup>2</jats:sup> − 1)<jats:sup><jats:italic>x</jats:italic></jats:sup> + ((<jats:italic>l</jats:italic> − <jats:italic>r</jats:italic>)<jats:italic>lm</jats:italic><jats:sup>2</jats:sup> + 1)<jats:sup><jats:italic>y</jats:italic></jats:sup> = (<jats:italic>lm</jats:italic>)<jats:sup><jats:italic>z</jats:italic></jats:sup> has only the positive integer solution (<jats:italic>x,y,z</jats:italic>) = (1<jats:italic>,</jats:italic>1<jats:italic>,</jats:italic>2).</jats:p>\",\"PeriodicalId\":51187,\"journal\":{\"name\":\"Studia Scientiarum Mathematicarum Hungarica\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2020-03-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studia Scientiarum Mathematicarum Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1556/012.2020.57.2.1457\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Scientiarum Mathematicarum Hungarica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1556/012.2020.57.2.1457","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
A note on the ternary purely exponential diophantine equation Ax + By = Cz with A + B = C2
Let l,m,r be fixed positive integers such that 2| l, 3lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm2 − 1,(l − r)lm2 + 1} > 30, then the equation (rlm2 − 1)x + ((l − r)lm2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).
期刊介绍：
The journal publishes original research papers on various fields of mathematics, e.g., algebra, algebraic geometry, analysis, combinatorics, dynamical systems, geometry, mathematical logic, mathematical statistics, number theory, probability theory, set theory, statistical physics and topology.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
