{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:54Z","timestamp":1776785814793,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,3,29]],"date-time":"2007-03-29T00:00:00Z","timestamp":1175126400000},"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                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an odd prime and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/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=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y squared equals x left-parenthesis x plus 2 Superscript a Baseline p Superscript b Baseline right-parenthesis left-parenthesis x minus 2 Superscript a Baseline p Superscript b Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>a<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>a<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">y^2 = x(x+2^ap^b)(x-2^ap^b)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be easily reduced to the resolution of the unit equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u plus StartRoot 2 EndRoot v equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u+\\sqrt {2}v = 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 2 EndRoot comma StartRoot p EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}(\\sqrt {2},\\sqrt {p})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The solutions of the latter equation are given by Wildanger\u2019s algorithm.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01841-2","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1585-1593","source":"Crossref","is-referenced-by-count":4,"title":["Practical solution of the Diophantine equation \ud835\udc66\u00b2=\ud835\udc65(\ud835\udc65+2^{\ud835\udc4e}\ud835\udc5d^{\ud835\udc4f})(\ud835\udc65-2^{\ud835\udc4e}\ud835\udc5d^{\ud835\udc4f})"],"prefix":"10.1090","volume":"75","author":[{"given":"Konstantinos","family":"Draziotis","sequence":"first","affiliation":[]},{"given":"Dimitrios","family":"Poulakis","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,29]]},"reference":[{"issue":"12","key":"1","doi-asserted-by":"publisher","first-page":"3481","DOI":"10.1090\/S0002-9939-99-05041-8","article-title":"The Diophantine equation \ud835\udc4f\u00b2\ud835\udc4b\u2074-\ud835\udc51\ud835\udc4c\u00b2=1","volume":"127","author":"Bennett, Michael A.","year":"1999","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1006\/jnth.1999.2430","article-title":"Integral points in arithmetic progression on \ud835\udc66\u00b2=\ud835\udc65(\ud835\udc65\u00b2-\ud835\udc5b\u00b2)","volume":"80","author":"Bremner, A.","year":"2000","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1017\/S0004972700031592","article-title":"On the size of integer solutions of elliptic equations","volume":"57","author":"Bugeaud, Yann","year":"1998","journal-title":"Bull. Austral. Math. 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