{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:45:49Z","timestamp":1776725149377,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"232","license":[{"start":{"date-parts":[[2001,2,21]],"date-time":"2001-02-21T00:00:00Z","timestamp":982713600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We propose an efficient search algorithm to solve the equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x cubed plus y cubed plus 2 z cubed equals n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x^3+y^3+ 2z^3=n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for a fixed value of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . By parametrizing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue z EndAbsoluteValue\">\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>z<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|z|<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , this algorithm obtains\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue x EndAbsoluteValue\">\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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|x|<\/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=\"StartAbsoluteValue y EndAbsoluteValue\">\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>y<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|y|<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (if they exist) by solving a quadratic equation derived from divisors of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 StartAbsoluteValue z EndAbsoluteValue cubed plus-or-minus n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">|<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2|z|^3 \\pm n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    below 1000 for which no solution had previously been found. We found three new integer solutions for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 183 comma 491\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>183<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mn>491<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=183, \\ 491<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and 931 in the range of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue z EndAbsoluteValue less-than-or-equal-to 5 dot 10 Superscript 7\">\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>z<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mn>7<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|z| \\le 5 \\cdot 10^7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01202-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1735-1742","source":"Crossref","is-referenced-by-count":3,"title":["On searching for solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+2\ud835\udc67\u00b3=\ud835\udc5b"],"prefix":"10.1090","volume":"69","author":[{"given":"Kenji","family":"Koyama","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2000,2,21]]},"reference":[{"key":"1","unstructured":"J. H. E. Cohn, private communication (1995)."},{"issue":"5","key":"2","first-page":"64","article-title":"Sums of four cubes","volume":"1966","author":"Dem\u2032janenko, V. A.","year":"1966","journal-title":"Izv. Vys\\v{s}. U\\v{c}ebn. Zaved. Matematika","ISSN":"https:\/\/id.crossref.org\/issn\/0021-3446","issn-type":"print"},{"key":"3","series-title":"Problem Books in Mathematics","isbn-type":"print","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0387905936"},{"key":"4","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3585-4","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387942890","edition":"2"},{"issue":"203","key":"5","doi-asserted-by":"publisher","first-page":"235","DOI":"10.2307\/2152950","article-title":"On solving the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc58 on a vector computer","volume":"61","author":"Heath-Brown, D. R.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"K. Koyama,  Tables of solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b, Math. Comp. 62 (1994), 941-942.","DOI":"10.1090\/S0025-5718-94-99733-0"},{"issue":"218","key":"7","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1090\/S0025-5718-97-00830-2","article-title":"On searching for solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b","volume":"66","author":"Koyama, Kenji","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","unstructured":"R. F. Lukes, private communication (1995)."},{"key":"9","series-title":"Pure and Applied Mathematics, Vol. 30","volume-title":"Diophantine equations","author":"Mordell, L. J.","year":"1969"},{"issue":"227","key":"10","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1090\/S0025-5718-99-01067-4","article-title":"Nonexistence conditions of a solution for the congruence \ud835\udc65^{\ud835\udc58}\u2081+\u22ef+\ud835\udc65^{\ud835\udc58}_{\ud835\udc60}\u2261\ud835\udc41 (mod \ud835\udc5d\u207f)","volume":"68","author":"Sekigawa, Hiroshi","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-00-01202-3\/S0025-5718-00-01202-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-00-01202-3\/S0025-5718-00-01202-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:27:30Z","timestamp":1776724050000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-00-01202-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,2,21]]},"references-count":10,"journal-issue":{"issue":"232","published-print":{"date-parts":[[2000,10]]}},"alternative-id":["S0025-5718-00-01202-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01202-3","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":[[2000,2,21]]}}}