{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:27:31Z","timestamp":1776864451360,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"268","license":[{"start":{"date-parts":[[2010,1,29]],"date-time":"2010-01-29T00:00:00Z","timestamp":1264723200000},"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":"<p>\n                    It has been believed that the continued fraction expansion of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis alpha comma beta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\alpha ,\\beta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 1 comma alpha comma beta\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(1,\\alpha ,\\beta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -basis of a real cubic field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:annotation encoding=\"application\/x-tex\">)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis mathematical left-angle RootIndex 3 StartRoot 3 EndRoot mathematical right-angle comma mathematical left-angle RootIndex 3 StartRoot 9 EndRoot mathematical right-angle right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e8\n                              \n                            <\/mml:mo>\n                            <mml:mroot>\n                              <mml:mn>3<\/mml:mn>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mroot>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e9\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e8\n                              \n                            <\/mml:mo>\n                            <mml:mroot>\n                              <mml:mn>9<\/mml:mn>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mroot>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e9\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\langle \\sqrt [3]{3}\\rangle , \\langle \\sqrt [3]{9}\\rangle )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"mathematical left-angle x mathematical right-angle\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e8\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e9\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\langle x\\rangle<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denoting the fractional part of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis alpha comma beta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\alpha ,\\beta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02217-0","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T10:39:02Z","timestamp":1246358342000},"page":"2209-2222","source":"Crossref","is-referenced-by-count":11,"title":["A new multidimensional continued fraction algorithm"],"prefix":"10.1090","volume":"78","author":[{"given":"Jun-ichi","family":"Tamura","sequence":"first","affiliation":[]},{"given":"Shin-ichi","family":"Yasutomi","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,29]]},"reference":[{"key":"1","series-title":"Lecture Notes in Mathematics, Vol. 207","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0069405","volume-title":"The Jacobi-Perron algorithm---Its theory and application","author":"Bernstein, Leon","year":"1971"},{"key":"2","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1002\/mana.19670340106","article-title":"Numerische Ergebnisse zum Jacobischen Kettenbruchalgorithmus in rein-kubischen Zahlk\u00f6rpern","volume":"34","author":"Elsner, Ludwig","year":"1967","journal-title":"Math. Nachr.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-584X","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1345","DOI":"10.1017\/S0143385700010063","article-title":"On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof","volume":"16","author":"Fujita, T.","year":"1996","journal-title":"Ergodic Theory Dynam. Systems","ISSN":"https:\/\/id.crossref.org\/issn\/0143-3857","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/S0022-314X(02)00076-8","article-title":"On simultaneous approximation to (\ud835\udefc,\ud835\udefc\u00b2) with \ud835\udefc\u00b3+\ud835\udc58\ud835\udefc-1=0","volume":"99","author":"Ito, Shunji","year":"2003","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"5","unstructured":"S. Ito and S. Yasutomi, On simultaneous approximation to certain periodic points related to modified Jacobi-Perron algorithm, to appear in Advanced Studies in Pure Mathematics."},{"key":"6","unstructured":"GiNaC web site: http:\/\/www.ginac.de\/."},{"key":"7","doi-asserted-by":"crossref","unstructured":"D.N. Lehmer, On Jacobi\u2019s Extension of the Continued Fraction Algorithm, Proc Natl Acad Sci U S A. 1918 December; 4(12): 360-364.","DOI":"10.1073\/pnas.4.12.360"},{"key":"8","doi-asserted-by":"crossref","DOI":"10.5948\/UPO9780883859261","volume-title":"Continued fractions","author":"Olds, C. D.","year":"1963"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01449880","article-title":"Grundlagen f\u00fcr eine Theorie des Jacobischen Kettenbruchalgorithmus","volume":"64","author":"Perron, Oskar","year":"1907","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"10","first-page":"184","article-title":"A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm","volume":"67","author":"Podsypanin, E. V.","year":"1977","journal-title":"Zap. Nau\\v{c}n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)"},{"key":"11","series-title":"Oxford Science Publications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198506867.001.0001","volume-title":"Multidimensional continued fractions","author":"Schweiger, Fritz","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0198506864"},{"key":"12","unstructured":"J. Tamura, A new approach to higher dimensional continued fractions, preprint."},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"301","DOI":"10.4064\/aa-71-4-301-329","article-title":"A class of transcendental numbers having explicit \ud835\udc54-adic and Jacobi-Perron expansions of arbitrary dimension","volume":"71","author":"Tamura, Jun-ichi","year":"1995","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/S0025-5718-09-02217-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/S0025-5718-09-02217-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:12:43Z","timestamp":1776787963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,29]]},"references-count":13,"journal-issue":{"issue":"268","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["S0025-5718-09-02217-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02217-0","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":[[2009,1,29]]}}}