{"title":"Distribution of Digits of Continued Fractions","authors":"","doi":"10.2307/j.ctv131bw89.15","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.15","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123754003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Appendix A.","authors":"","doi":"10.2307/j.ctv131bw89.24","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.24","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"36 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131211500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this article, we investigate the solutions of the Diophantine equations p x + ( p + 1) y + ( p + 2) z = M 2 for primes p ≥ 2 when 1 ≤ x , y , z ≤ 2. We establish : (i) When p = 2 and x = y = z = 1, the equation has a unique solution. (ii) When p = 4 N + 1 and 1 ≤ x , y , z ≤ 2, the equations have no solutions. (iii) When p = 4 N + 3 and x = y = z = 1, the equation has infinitely many solutions. (iv) When 3 ≤ p ≤ 199 and x = 1, y = z = 2, the equation has exactly one solution. (v) In all other cases 1 ≤ x , y , z ≤ 2 which are not mentioned above, the equations have no solutions.
． 本文研究了当1≤x, y, z≤2时，当质数p≥2时，Diophantine方程p x + (p + 1) y + (p + 2) z = m2的解。我们建立:(i)当p = 2且x = y = z = 1时，方程有唯一解。(ii)当p = 4n + 1且1≤x, y, z≤2时，方程无解。(iii)当p = 4 N + 3, x = y = z = 1时，方程有无穷多个解。(iv)当3≤p≤199,x = 1, y = z = 2时，方程只有一个解。(v)除上述情况外，1≤x, y, z≤2时，方程无解。
{"title":"Solutions to Diophantine Equations","authors":"","doi":"10.2307/j.ctv131bw89.9","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.9","url":null,"abstract":". In this article, we investigate the solutions of the Diophantine equations p x + ( p + 1) y + ( p + 2) z = M 2 for primes p ≥ 2 when 1 ≤ x , y , z ≤ 2. We establish : (i) When p = 2 and x = y = z = 1, the equation has a unique solution. (ii) When p = 4 N + 1 and 1 ≤ x , y , z ≤ 2, the equations have no solutions. (iii) When p = 4 N + 3 and x = y = z = 1, the equation has infinitely many solutions. (iv) When 3 ≤ p ≤ 199 and x = 1, y = z = 2, the equation has exactly one solution. (v) In all other cases 1 ≤ x , y , z ≤ 2 which are not mentioned above, the equations have no solutions.","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124912751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"From Nuclear Physics to L-Functions","authors":"","doi":"10.2307/j.ctv131bw89.20","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.20","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127662982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Introduction to the Circle Method","authors":"M. Murty, Kaneenika Sinha","doi":"10.2307/j.ctv131bw89.18","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.18","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128031634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Explicit Formula and Density Conjectures","authors":"","doi":"10.2307/j.ctv131bw89.23","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.23","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115550787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Arithmetic Functions","authors":"J. Conrey","doi":"10.2307/j.ctv131bw89.7","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.7","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125792417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"{nkα} and Poissonian Behavior","authors":"","doi":"10.2307/j.ctv131bw89.17","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.17","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126838812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Proof of Roth’s Theorem","authors":"","doi":"10.2307/j.ctv131bw89.11","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.11","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122132341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Zeta and L-Functions","authors":"","doi":"10.2307/j.ctv131bw89.8","DOIUrl":"https://doi.org/10.2307/j.ctv131bw89.8","url":null,"abstract":"","PeriodicalId":281730,"journal":{"name":"An Invitation to Modern Number Theory","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132165665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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