{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:47Z","timestamp":1776785807099,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,4,6]],"date-time":"2007-04-06T00:00:00Z","timestamp":1175817600000},"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 completely solve diophantine equations of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Y squared equals upper X cubed plus-or-minus p Superscript k Baseline upper X comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>Y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Y^{2}=X^{3}\\pm p^{k}X,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a positive integer, using a reduction to some quartic elliptic equations, which can be solved with well known methods.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01852-7","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1493-1505","source":"Crossref","is-referenced-by-count":6,"title":["Integer points on the curve \ud835\udc4c\u00b2=\ud835\udc4b\u00b3\u00b1\ud835\udc5d^{\ud835\udc58}\ud835\udc4b"],"prefix":"10.1090","volume":"75","author":[{"given":"Konstantinos","family":"Draziotis","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,4,6]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1090\/S0002-9947-00-02658-1","article-title":"On the representation of unity by binary cubic forms","volume":"353","author":"Bennett, Michael A.","year":"2001","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1006\/jnth.1997.2018","article-title":"Complete solution of the Diophantine equation \ud835\udc4b\u00b2+1=\ud835\udc51\ud835\udc4c\u2074 and a related family of quartic Thue equations","volume":"62","author":"Chen, Jian Hua","year":"1997","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":"180","DOI":"10.7146\/math.scand.a-11746","article-title":"The Diophantine equation \ud835\udc66\u00b2=\ud835\udc37\ud835\udc65\u2074+1. III","volume":"42","author":"Cohn, J. H. E.","year":"1978","journal-title":"Math. Scand.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5521","issn-type":"print"},{"issue":"219","key":"4","doi-asserted-by":"publisher","first-page":"1347","DOI":"10.1090\/S0025-5718-97-00851-X","article-title":"The Diophantine equation \ud835\udc65\u2074+1=\ud835\udc37\ud835\udc66\u00b2","volume":"66","author":"Cohn, J. H. E.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1112\/jlms\/s2-40.3.385","article-title":"On heterogeneous spaces","volume":"40","author":"Coombes, Kevin R.","year":"1989","journal-title":"J. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"62","key":"6","first-page":"iv+143","article-title":"Minorations de formes lin\u00e9aires de logarithmes elliptiques","author":"David, Sinnou","year":"1995","journal-title":"M\\'{e}m. Soc. Math. France (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"171","DOI":"10.4064\/aa-68-2-171-192","article-title":"Computing integral points on elliptic curves","volume":"68","author":"Gebel, J.","year":"1994","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"8","unstructured":"Genocchi, Sur l\u2019impossibilite de quelques egalites doubles, C. R. Acad.Sci. Paris, 78 (1874), 423-436."},{"issue":"6","key":"9","first-page":"91","article-title":"The negative Pell equation and Pythagorean triples","volume":"76","author":"Grytczuk, Aleksander","year":"2000","journal-title":"Proc. Japan Acad. Ser. A Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0386-2194","issn-type":"print"},{"key":"10","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-07010-9","volume-title":"Elliptic curves: Diophantine analysis","volume":"231","author":"Lang, Serge","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/3540084894"},{"issue":"5","key":"11","first-page":"27","article-title":"Zur Theorie der Gleichung \ud835\udc65\u00b2+1=\ud835\udc37\ud835\udc66\u2074","volume":"1942","author":"Ljunggren, Wilhelm","year":"1942","journal-title":"Avh. Norske Vid.-Akad. Oslo I","ISSN":"https:\/\/id.crossref.org\/issn\/0549-7019","issn-type":"print"},{"key":"12","unstructured":"\\bysame, Einige Eigenschften der Einheiten reel Quadratischer und rein-bi-quadratischer Zahlkorper. Skr. Norske Vid. Akad. Oslo I, v.1936, no.12."},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1216\/rmjm\/1020171571","article-title":"A generalization of a theorem of Cohn on the equation \ud835\udc65\u00b3-\ud835\udc41\ud835\udc66\u00b2=\u00b11","volume":"31","author":"Luca, F.","year":"2001","journal-title":"Rocky Mountain J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0035-7596","issn-type":"print"},{"key":"14","series-title":"Pure and Applied Mathematics, Vol. 30","volume-title":"Diophantine equations","author":"Mordell, L. J.","year":"1969"},{"key":"15","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1112\/jlms\/s1-39.1.161","article-title":"The Diophantine equation \ud835\udc66\u00b2=\ud835\udc37\ud835\udc65\u2074+1","volume":"39","author":"Mordell, L. J.","year":"1964","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1007\/s000170050053","article-title":"A simple method for solving the Diophantine equation \ud835\udc4c\u00b2=\ud835\udc4b\u2074+\ud835\udc4e\ud835\udc4b\u00b3+\ud835\udc4f\ud835\udc4b\u00b2+\ud835\udc50\ud835\udc4b+\ud835\udc51","volume":"54","author":"Poulakis, Dimitrios","year":"1999","journal-title":"Elem. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0013-6018","issn-type":"print"},{"issue":"2","key":"17","first-page":"54","article-title":"A note on the Diophantine equation \ud835\udc4b\u00b2-\ud835\udc51\ud835\udc4c\u2074=1 with prime discriminant","volume":"27","author":"Poulakis, D.","year":"2005","journal-title":"C. R. Math. Acad. Sci. Soc. R. Can.","ISSN":"https:\/\/id.crossref.org\/issn\/0706-1994","issn-type":"print"},{"key":"18","series-title":"Oxford Science Publications","isbn-type":"print","volume-title":"A course in number theory","author":"Rose, H. E.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0198534795","edition":"2"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"629","DOI":"10.5802\/jtnb.379","article-title":"R\u00e9sultats \u00e9l\u00e9mentaires sur certaines \u00e9quations diophantiennes","volume":"14","author":"Samuel, Pierre","year":"2002","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"key":"20","doi-asserted-by":"publisher","first-page":"185","DOI":"10.4064\/aa-4-3-185-208","article-title":"Sur certaines hypoth\u00e8ses concernant les nombres premiers","volume":"4","author":"Schinzel, A.","year":"1958","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"21","series-title":"North-Holland Mathematical Library","isbn-type":"print","volume-title":"Elementary theory of numbers","volume":"31","author":"Sierpi\u0144ski, W.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0444866620","edition":"2"},{"key":"22","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1920-8","volume-title":"The arithmetic of elliptic curves","volume":"106","author":"Silverman, Joseph H.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0387962034"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1017\/S0305004100072698","article-title":"\ud835\udc46-integral points on elliptic curves","volume":"116","author":"Smart, N. P.","year":"1994","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"issue":"2","key":"24","doi-asserted-by":"publisher","first-page":"177","DOI":"10.4064\/aa-67-2-177-196","article-title":"Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms","volume":"67","author":"Stroeker, R. J.","year":"1994","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"25","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1080\/10586458.1999.10504395","article-title":"On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement","volume":"8","author":"Stroeker, Roel J.","year":"1999","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"39","DOI":"10.4064\/aa120-1-3","article-title":"Solving a family of Thue equations with an application to the equation \ud835\udc65\u00b2-\ud835\udc37\ud835\udc66\u2074=1","volume":"120","author":"Togbe, A.","year":"2005","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/0022-314X(89)90014-0","article-title":"On the practical solution of the Thue equation","volume":"31","author":"Tzanakis, N.","year":"1989","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"28","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0072989","volume-title":"Diophantine approximations and value distribution theory","volume":"1239","author":"Vojta, Paul","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540175512"},{"key":"29","isbn-type":"print","first-page":"531","article-title":"Diophantine equations of the form \ud835\udc4e\ud835\udc4b\u2074-\ud835\udc4f\ud835\udc4c\u00b2=\u00b11","author":"Walsh, P. G.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3110163047"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s000130050376","article-title":"A note on a theorem of Ljunggren and the Diophantine equations \ud835\udc65\u00b2-\ud835\udc58\ud835\udc65\ud835\udc66\u00b2+\ud835\udc66\u2074=1,4","volume":"73","author":"Walsh, Gary","year":"1999","journal-title":"Arch. Math. (Basel)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-889X","issn-type":"print"},{"issue":"177","key":"31","doi-asserted-by":"publisher","first-page":"425","DOI":"10.2307\/2007900","article-title":"Large integral points on elliptic curves","volume":"48","author":"Zagier, Don","year":"1987","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\/2006-75-255\/S0025-5718-06-01852-7\/S0025-5718-06-01852-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01852-7\/S0025-5718-06-01852-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:36:49Z","timestamp":1776782209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01852-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,4,6]]},"references-count":31,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01852-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01852-7","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":[[2006,4,6]]}}}