{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:43:44Z","timestamp":1776789824265,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,8,12]],"date-time":"2009-08-12T00:00:00Z","timestamp":1250035200000},"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                    Given positive integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a comma b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a,b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to compute a generating system for the numerical semigroup whose elements are all positive integer solutions of the inequality\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a x bold m bold o bold d b less-than-or-equal-to c x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">m<\/mml:mi>\n                              <mml:mi mathvariant=\"bold\">o<\/mml:mi>\n                              <mml:mi mathvariant=\"bold\">d<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a x \\,\\mathbf {mod}\\, b\\leq cx<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is equivalent to computing a B\u00e9zout sequence connecting two reduced fractions. We prove that a proper B\u00e9zout sequence is completely determined by its ends and we give an algorithm to compute the unique proper B\u00e9zout sequence connecting two reduced fractions. We also relate B\u00e9zout sequences with paths in the Stern-Brocot tree and use this tree to compute the minimal positive integer solution of the above inequality.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02173-x","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"1211-1226","source":"Crossref","is-referenced-by-count":8,"title":["Proportionally modular diophantine inequalities and the Stern-Brocot tree"],"prefix":"10.1090","volume":"78","author":[{"given":"M.","family":"Bullejos","sequence":"first","affiliation":[]},{"given":"J.","family":"Rosales","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,8,12]]},"reference":[{"key":"1","unstructured":"A. Bogomolny, Stern-Brocot Tree. Binary Encoding; see the website: www.cut-the-knot.org\/blue\/encoding.shtml."},{"key":"2","unstructured":"R. Graham, D. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, 1994."},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/BF01300131","article-title":"Complexity of the Frobenius problem","volume":"16","author":"Ram\u00edrez-Alfons\u00edn, J. L.","year":"1996","journal-title":"Combinatorica","ISSN":"https:\/\/id.crossref.org\/issn\/0209-9683","issn-type":"print"},{"key":"4","series-title":"Oxford Lecture Series in Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198568209.001.0001","volume-title":"The Diophantine Frobenius problem","volume":"30","author":"Ram\u00edrez Alfons\u00edn, J. L.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780198568209"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.jnt.2003.06.002","article-title":"Proportionally modular Diophantine inequalities","volume":"103","author":"Rosales, J. C.","year":"2003","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"J. C. Rosales, P. A. Garc\u00eda-S\u00e1nchez and J. M. Urbano-Blanco, The set of solutions of a proportionally modular diophantine inequality, Journal of Number Theory 128 (2008), 453-467.","DOI":"10.1016\/j.jnt.2007.11.002"},{"key":"7","doi-asserted-by":"crossref","unstructured":"J. C. Rosales and P. Vasco, The smallest positive integer that is a solution of a proportionally modular diophantine inequality, to appear in Mathematical Inequalities & Applications (2008), www.ele-math.com.","DOI":"10.7153\/mia-11-14"},{"key":"8","unstructured":"J. J. Sylvester, Mathematical questions with their solutions, Educational Times 41 (1884), 21."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02173-X\/S0025-5718-08-02173-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02173-X\/S0025-5718-08-02173-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:59:09Z","timestamp":1776787149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02173-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,8,12]]},"references-count":8,"journal-issue":{"issue":"266","published-print":{"date-parts":[[2009,4]]}},"alternative-id":["S0025-5718-08-02173-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02173-x","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":[[2008,8,12]]}}}