{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:48:39Z","timestamp":1776721719215,"version":"3.51.2"},"reference-count":5,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R\">\n                        <mml:semantics>\n                          <mml:mi>R<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">R<\/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=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be positive integers with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R greater-than upper S\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R&gt;S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We shall call the simultaneous Diophantine equations\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row 1st Column x squared minus upper R y squared 2nd Column a m p semicolon equals 1 comma z squared minus upper S y squared 3rd Column a m p semicolon equals 1 EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" side=\"left\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>R<\/mml:mi>\n                                <mml:msup>\n                                  <mml:mi>y<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                              <\/mml:mtd>\n                              <mml:mtd>\n                                <mml:mi>a<\/mml:mi>\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>;<\/mml:mo>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mtext>\u00a0<\/mml:mtext>\n                                <mml:msup>\n                                  <mml:mi>z<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>S<\/mml:mi>\n                                <mml:msup>\n                                  <mml:mi>y<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                              <\/mml:mtd>\n                              <mml:mtd>\n                                <mml:mi>a<\/mml:mi>\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>;<\/mml:mo>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{align*} x^2-Ry^2&amp;=1,\\ z^2-Sy^2&amp;=1 \\end{align*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    <italic>\n                      simultaneous Pell equations in\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R\">\n                          <mml:semantics>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:annotation encoding=\"application\/x-tex\">R<\/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=\"upper S\">\n                          <mml:semantics>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      .\n                    <\/italic>\n                    Each such pair has the trivial solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 1 comma 0 comma 1 right-parenthesis\">\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:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(1,0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    but some pairs have nontrivial solutions too. For example, if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R equals 11\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>11<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R=11<\/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=\"upper S equals 56\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>56<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S=56<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 199 comma 60 comma 449 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>199<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>60<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>449<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(199, 60, 449)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R greater-than upper S less-than-or-equal-to 200\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>200<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R&gt;S\\le 200<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00687-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"355-359","source":"Crossref","is-referenced-by-count":14,"title":["Simultaneous Pell equations"],"prefix":"10.1090","volume":"65","author":[{"given":"W.","family":"Anglin","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1093\/qmath\/20.1.129","article-title":"The equations 3\ud835\udc65\u00b2-2=\ud835\udc66\u00b2 and 8\ud835\udc65\u00b2-7=\ud835\udc67\u00b2","volume":"20","author":"Baker, A.","year":"1969","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"2","isbn-type":"print","volume-title":"An introduction to the theory of numbers","author":"Niven, Ivan","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0471625469","edition":"5"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1017\/S0305004100064598","article-title":"Simultaneous Pellian equations","volume":"103","author":"Pinch, R. G. E.","year":"1988","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"key":"4","unstructured":"C. L. Siegel, \u00dcber einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929."},{"key":"5","doi-asserted-by":"publisher","first-page":"257","DOI":"10.4064\/aa-37-1-257-283","article-title":"A lower bound for linear forms in logarithms","volume":"37","author":"Waldschmidt, Michel","year":"1980","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\/1996-65-213\/S0025-5718-96-00687-4\/S0025-5718-96-00687-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00687-4\/S0025-5718-96-00687-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:02:29Z","timestamp":1776718949000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00687-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":5,"journal-issue":{"issue":"213","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["S0025-5718-96-00687-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00687-4","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":[[1996]]}}}