{"title":"关于p-adic连分式算法的周期性","authors":"Nadir Murru, Giuliano Romeo, Giordano Santilli","doi":"10.1007/s10231-023-01347-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), for generating continued fractions in the field of <i>p</i>-adic numbers <span>\\(\\mathbb Q_p\\)</span>. First of all, we obtain an analogue of the Galois’ Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in <span>\\(\\mathbb Q_p\\)</span> that have a periodic expansion with period of length 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).\n</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-023-01347-6.pdf","citationCount":"6","resultStr":"{\"title\":\"On the periodicity of an algorithm for p-adic continued fractions\",\"authors\":\"Nadir Murru, Giuliano Romeo, Giordano Santilli\",\"doi\":\"10.1007/s10231-023-01347-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), for generating continued fractions in the field of <i>p</i>-adic numbers <span>\\\\(\\\\mathbb Q_p\\\\)</span>. First of all, we obtain an analogue of the Galois’ Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in <span>\\\\(\\\\mathbb Q_p\\\\)</span> that have a periodic expansion with period of length 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).\\n</p></div>\",\"PeriodicalId\":8265,\"journal\":{\"name\":\"Annali di Matematica Pura ed Applicata\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10231-023-01347-6.pdf\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annali di Matematica Pura ed Applicata\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10231-023-01347-6\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-023-01347-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the periodicity of an algorithm for p-adic continued fractions
In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), for generating continued fractions in the field of p-adic numbers \(\mathbb Q_p\). First of all, we obtain an analogue of the Galois’ Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in \(\mathbb Q_p\) that have a periodic expansion with period of length 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.