{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:02:18Z","timestamp":1776830538170,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We propose a new 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 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: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+z^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 x EndAbsoluteValue equals\">\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                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|x|=<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    min\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis StartAbsoluteValue x EndAbsoluteValue comma StartAbsoluteValue y EndAbsoluteValue comma StartAbsoluteValue z EndAbsoluteValue right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\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                            <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:mo>,<\/mml:mo>\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 stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(|x|, |y|, |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 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                    and\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                    (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=\"StartAbsoluteValue x EndAbsoluteValue cubed plus-or-minus n\">\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: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\">|x|^3 \\pm n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . By using several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for 51 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 (except\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n identical-to plus-or-minus 4 left-parenthesis mod 9 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>mod<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mn>9<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\equiv \\pm 4 (\\operatorname {mod}9)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) for which no solution has previously been found. We found eight 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 75 comma 435 comma 444 comma 501 comma 600 comma 618 comma 912 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>75<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>435<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>444<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>501<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>600<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>618<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>912<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=75, 435, 444, 501, 600, 618, 912,<\/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=\"969\">\n                        <mml:semantics>\n                          <mml:mn>969<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">969<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    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 x EndAbsoluteValue less-than-or-equal-to 2 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>x<\/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>2<\/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\">|x| \\le 2 \\cdot 10^7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00830-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"841-851","source":"Crossref","is-referenced-by-count":9,"title":["On searching for solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b"],"prefix":"10.1090","volume":"66","author":[{"given":"Kenji","family":"Koyama","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yukio","family":"Tsuruoka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiroshi","family":"Sekigawa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","isbn-type":"print","first-page":"87","article-title":"On sums of three cubes","author":"Bremner, Andrew","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0821803123"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1090\/conm\/166\/01628","article-title":"On sums of three integral cubes","author":"Conn, W.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/082185173X"},{"key":"3","doi-asserted-by":"publisher","first-page":"408","DOI":"10.2307\/2003763","article-title":"Solutions of the diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3=\ud835\udc67\u00b3-\ud835\udc51","volume":"18","author":"Gardiner, V. L.","year":"1964","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","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":"5","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":"6","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":"7","unstructured":"W. C. Jagy, Progress report, private communication, January 1995."},{"key":"8","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"},{"key":"9","unstructured":"K. Koyama, On the solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b, Trans. of Inst. of Electronics, Information and Communication Engineers (IEICE in Japan), Vol.E78-A, No. 3 (1995), 444-449."},{"key":"10","unstructured":"R. F. Lukes, A very fast electronic number sieve, Ph. D. Thesis, Univ. of Manitoba (1995)."},{"key":"11","doi-asserted-by":"publisher","first-page":"611","DOI":"10.2307\/1968946","article-title":"On Frobeniusean algebras. I","volume":"40","author":"Nakayama, Tadasi","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"12","series-title":"Pure and Applied Mathematics, Vol. 30","volume-title":"Diophantine equations","author":"Mordell, L. J.","year":"1969"},{"issue":"4","key":"13","first-page":"285","article-title":"Computational number theory at CWI in 1970\u20131994","volume":"7","author":"te Riele, H. J. J.","year":"1994","journal-title":"CWI Quarterly","ISSN":"https:\/\/id.crossref.org\/issn\/0922-5366","issn-type":"print"},{"key":"14","unstructured":"H. Sekigawa and K. Koyama, Existence condition of solutions of congruence \ud835\udc65\u207f+\ud835\udc66\u207f\u2261\ud835\udc5a(gi^j^{_}), in preparation."},{"issue":"165","key":"15","doi-asserted-by":"publisher","first-page":"235","DOI":"10.2307\/2007575","article-title":"A note on the Diophantine equation \ud835\udc65\u207f+\ud835\udc66\u207f+\ud835\udc67\u207f=3","volume":"42","author":"Scarowsky, Manny","year":"1984","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\/1997-66-218\/S0025-5718-97-00830-2\/S0025-5718-97-00830-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00830-2\/S0025-5718-97-00830-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:27:40Z","timestamp":1776720460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00830-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"references-count":15,"journal-issue":{"issue":"218","published-print":{"date-parts":[[1997,4]]}},"alternative-id":["S0025-5718-97-00830-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-97-00830-2","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":[[1997]]}}}