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\n We\u00a0report on our search for solutions of the Diophantine equation\n \n \n \n \n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n y<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n z<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n =<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n x^3 + y^3 + z^3 = n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n n<\/mml:mi>\n ><\/mml:mo>\n 1000<\/mml:mn>\n <\/mml:mrow>\n n > 1000<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\u00a0\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n x<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n y<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n z<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n \n 10<\/mml:mn>\n \n 14<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n |x|, |y|, |z| > 10^{14}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-08-02168-6","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"1227-1230","source":"Crossref","is-referenced-by-count":7,"title":["New sums of three cubes"],"prefix":"10.1090","volume":"78","author":[{"given":"Andreas-Stephan","family":"Elsenhans","sequence":"first","affiliation":[]},{"given":"J\u00f6rg","family":"Jahnel","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,8,20]]},"reference":[{"key":"1","unstructured":"[AMD] Software Optimization Guide for AMD Athlon^{\ud835\udc47\ud835\udc40} 64 and AMD Opteron^{\ud835\udc47\ud835\udc40} Processors, Rev. 3.04, AMD, Sunnyvale, CA, 2004."},{"issue":"259","key":"2","doi-asserted-by":"publisher","first-page":"1683","DOI":"10.1090\/S0025-5718-07-01947-3","article-title":"New integer representations as the sum of three cubes","volume":"76","author":"Beck, Michael","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","unstructured":"[Be] Bernstein, D.: Threecubes, available at: http:\/\/cr.yp.to\/threecubes.html."},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/10722028_2","article-title":"Rational points near curves and small nonzero |\ud835\udc65\u00b3-\ud835\udc66\u00b2| via lattice reduction","author":"Elkies, Noam D.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540676953"},{"key":"5","unstructured":"[EJ] Elsenhans, A.-S. and Jahnel, J.: List of solutions of \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b for \ud835\udc5b<1000 neither a cube nor twice a cube, available at: http:\/\/www.uni-math.gwdg.de\/jahnel\/ threecubes_20070419.txt."},{"issue":"170","key":"6","doi-asserted-by":"publisher","first-page":"463","DOI":"10.2307\/2007966","article-title":"Improved methods for calculating vectors of short length in a lattice, including a complexity analysis","volume":"44","author":"Fincke, U.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"200","key":"7","doi-asserted-by":"publisher","first-page":"613","DOI":"10.2307\/2153078","article-title":"The density of zeros of forms for which weak approximation fails","volume":"59","author":"Heath-Brown, D. R.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"203","key":"8","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"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02168-6\/S0025-5718-08-02168-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02168-6\/S0025-5718-08-02168-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:59:11Z","timestamp":1776787151000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02168-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,8,20]]},"references-count":8,"journal-issue":{"issue":"266","published-print":{"date-parts":[[2009,4]]}},"alternative-id":["S0025-5718-08-02168-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02168-6","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":[[2008,8,20]]}}}