{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:23:41Z","timestamp":1776795821494,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"282","license":[{"start":{"date-parts":[[2013,10,15]],"date-time":"2013-10-15T00:00:00Z","timestamp":1381795200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We discuss continued fractions on real quadratic number fields of class number\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . If the field has the property of being\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions. Although it is conjectured that all real quadratic fields of class number\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stage euclidean, this property has been proven for only a few of them. The main result of this paper is an algorithm that, given a real quadratic field of class number\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , verifies this conjecture, and produces as byproduct enough data to efficiently compute continued fraction expansions. If the field was not\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stage euclidean, then the algorithm would not terminate. As an application, we enlarge the list of known\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stage euclidean fields, by proving that all real quadratic fields of class number\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and discriminant less than\n \n \n \n 8000<\/mml:mn>\n 8000<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stage euclidean.\n <\/p>","DOI":"10.1090\/s0025-5718-2012-02620-2","type":"journal-article","created":{"date-parts":[[2012,10,15]],"date-time":"2012-10-15T13:37:29Z","timestamp":1350308249000},"page":"1223-1233","source":"Crossref","is-referenced-by-count":2,"title":["Continued fractions in 2-stage Euclidean quadratic fields"],"prefix":"10.1090","volume":"82","author":[{"given":"Xavier","family":"Guitart","sequence":"first","affiliation":[]},{"given":"Marc","family":"Masdeu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2012,10,15]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1515\/crll.1976.282.133","article-title":"A weakening of the Euclidean property for integral domains and applications to algebraic number theory. I","volume":"282","author":"Cooke, George E.","year":"1976","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1515\/crll.1976.283-284.71","article-title":"A weakening of the Euclidean property for integral domains and applications to algebraic number theory. II","volume":"283(284)","author":"Cooke, George E.","year":"1976","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1080\/00927877508822057","article-title":"On the construction of division chains in algebraic number rings, with applications to \ud835\udc46\ud835\udc3f\u2082","volume":"3","author":"Cooke, George","year":"1975","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"3","key":"4","first-page":"275","article-title":"Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields","volume":"51","author":"Cremona, J. E.","year":"1984","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"40","key":"5","doi-asserted-by":"publisher","first-page":"2153","DOI":"10.1155\/S1073792803131108","article-title":"Periods of Hilbert modular forms and rational points on elliptic curves","author":"Darmon, Henri","year":"2003","journal-title":"Int. Math. Res. Not.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1112\/plms\/s2-53.1.65","article-title":"Indefinite binary quadratic forms, and Euclid\u2019s algorithm in real quadratic fields","volume":"53","author":"Davenport, H.","year":"1951","journal-title":"Proc. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1080\/10586458.2008.10128875","article-title":"An algorithm for modular elliptic curves over real quadratic fields","volume":"17","author":"Demb\u00e9l\u00e9, Lassina","year":"2008","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"8","unstructured":"L. Demb\u00e9l\u00e9, J. Voight, Explicit methods for Hilbert modular forms. To appear in \u201cElliptic Curves, Hilbert modular forms and Galois deformations\u201d. arXiv:1010.5727v2."},{"key":"9","first-page":"58","article-title":"On the first inhomogeneous minimum of indefinite binary quadratic forms and Euclid\u2019s algorithm in real quadratic fields","volume":"28","author":"Ennola, Veikko","year":"1958","journal-title":"Ann. Univ. Turku. Ser. A I","ISSN":"https:\/\/id.crossref.org\/issn\/0082-7002","issn-type":"print"},{"key":"10","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199219858.001.0001","volume-title":"An introduction to the theory of numbers","author":"Hardy, G. H.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780199219865","edition":"6"},{"issue":"5","key":"11","first-page":"385","article-title":"The Euclidean algorithm in algebraic number fields","volume":"13","author":"Lemmermeyer, Franz","year":"1995","journal-title":"Exposition. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0723-0869","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2013-82-282\/S0025-5718-2012-02620-2\/S0025-5718-2012-02620-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-282\/S0025-5718-2012-02620-2\/S0025-5718-2012-02620-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:30:55Z","timestamp":1776792655000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-282\/S0025-5718-2012-02620-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,15]]},"references-count":11,"journal-issue":{"issue":"282","published-print":{"date-parts":[[2013,4]]}},"alternative-id":["S0025-5718-2012-02620-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2012-02620-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2012,10,15]]}}}