{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:03:59Z","timestamp":1776841439462,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,19]],"date-time":"2006-12-19T00:00:00Z","timestamp":1166486400000},"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                    We show that, within the hypercube\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue x EndAbsoluteValue comma StartAbsoluteValue y EndAbsoluteValue comma StartAbsoluteValue z EndAbsoluteValue comma StartAbsoluteValue w EndAbsoluteValue less-than-or-equal-to 2.5 dot 10 Superscript 6\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n  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10^6<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the Diophantine equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x Superscript 4 Baseline plus 2 y Superscript 4 Baseline equals z Superscript 4 Baseline plus 4 w Superscript 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x^4 + 2 y^4 = z^4 + 4 w^4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    admits essentially one and only one nontrivial solution, namely\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis plus-or-minus 1 484 801 comma plus-or-minus 1 203 120 comma plus-or-minus 1 169 407 comma plus-or-minus 1 157 520 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>484<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>801<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>203<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>120<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>169<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>407<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>157<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>520<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\pm 1\\,484\\,801, \\pm 1\\,203\\,120, \\pm 1\\,169\\,407, \\pm 1\\,157\\,520)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The\u00a0investigation is based on a systematic search by computer.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01805-3","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"935-940","source":"Crossref","is-referenced-by-count":5,"title":["The Diophantine Equation \ud835\udc65\u2074+2\ud835\udc66\u2074=\ud835\udc67\u2074+4\ud835\udc64\u2074"],"prefix":"10.1090","volume":"75","author":[{"given":"Andreas-Stephan","family":"Elsenhans","sequence":"first","affiliation":[]},{"given":"J\u00f6rg","family":"Jahnel","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,12,19]]},"reference":[{"key":"1","series-title":"London Mathematical Society Lecture Note Series","isbn-type":"print","volume-title":"Complex algebraic surfaces","volume":"68","author":"Beauville, Arnaud","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0521288150"},{"issue":"233","key":"2","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1090\/S0025-5718-00-01219-9","article-title":"Enumerating solutions to \ud835\udc5d(\ud835\udc4e)+\ud835\udc5e(\ud835\udc4f)=\ud835\udc5f(\ud835\udc50)+\ud835\udc60(\ud835\udc51)","volume":"70","author":"Bernstein, Daniel J.","year":"2001","journal-title":"Math. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[Fo] Forster, O.: Algorithmische Zahlentheorie, Vieweg, Braunschweig 1996.","DOI":"10.1007\/978-3-663-09239-1"},{"key":"6","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8170-8","volume-title":"Arithmetic of higher-dimensional algebraic varieties","volume":"226","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/081763259X"},{"key":"7","series-title":"Addison-Wesley Series in Computer Science","isbn-type":"print","volume-title":"Algorithms","author":"Sedgewick, Robert","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0201066726"},{"key":"8","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107359994","volume-title":"The algorithmic resolution of Diophantine equations","volume":"41","author":"Smart, Nigel P.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0521646332"},{"key":"9","unstructured":"[S-D] Swinnerton-Dyer, Sir P.: Rational points on fibered surfaces, in: Tschinkel, Y. (ed.): Mathematisches Institut, Seminars 2004, Universit\u00e4tsverlag, G\u00f6ttingen 2004, 103\u2013109."},{"key":"10","series-title":"Publications de l'Institut de Math\\'{e}matiques de l'Universit\\'{e} de Strasbourg [Publications of the Mathematical Institute of the University of Strasbourg]","volume-title":"Sur les courbes alg\\'{e}briques et les vari\\'{e}t\\'{e}s qui s'en d\\'{e}duisent","volume":"7","author":"Weil, Andr\u00e9","year":"1948"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01805-3\/S0025-5718-05-01805-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01805-3\/S0025-5718-05-01805-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:35:40Z","timestamp":1776782140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01805-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,12,19]]},"references-count":10,"journal-issue":{"issue":"254","published-print":{"date-parts":[[2006,4]]}},"alternative-id":["S0025-5718-05-01805-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01805-3","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":[[2005,12,19]]}}}