{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:08:59Z","timestamp":1776766139876,"version":"3.51.2"},"reference-count":38,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n If\n \n \n \n \n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n <\/mml:mrow>\n a, b<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are positive integers with\n \n \n \n \n b<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n a<\/mml:mi>\n <\/mml:mrow>\n b \\geq a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n n<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n n \\geq 3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then the equation of the title possesses at most one solution in positive integers\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n y<\/mml:mi>\n y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with the possible exceptions of\n \n \n \n \n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n ( a, b, n )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfying\n \n \n \n \n b<\/mml:mi>\n =<\/mml:mo>\n a<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n b = a + 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n 2<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n a<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n min<\/mml:mo>\n {<\/mml:mo>\n 0.3<\/mml:mn>\n n<\/mml:mi>\n ,<\/mml:mo>\n 83<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n 2 \\leq a \\leq \\min \\{ 0.3 n, 83 \\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n 17<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 347<\/mml:mn>\n <\/mml:mrow>\n 17 \\leq n \\leq 347<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00900-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"413-438","source":"Crossref","is-referenced-by-count":20,"title":["On the Diophantine equation |\ud835\udc4e\ud835\udc65\u207f-\ud835\udc4f\ud835\udc66\u207f|=1"],"prefix":"10.1090","volume":"67","author":[{"given":"Michael","family":"Bennett","sequence":"first","affiliation":[]},{"given":"Benjamin","family":"de Weger","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1112\/plms\/s3-14.3.385","article-title":"Rational approximations to certain algebraic numbers","volume":"14","author":"Baker, A.","year":"1964","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1093\/qmath\/15.1.375","article-title":"Rational approximations to \\root3\\of2 and other algebraic numbers","volume":"15","author":"Baker, A.","year":"1964","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1098\/rsta.1968.0010","article-title":"Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms","volume":"263","author":"Baker, A.","year":"1967","journal-title":"Philos. Trans. Roy. Soc. London Ser. 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Evertse, Upper Bounds for the Numbers of Solutions of Diophantine Equations, PhD Thesis, Leiden, 1983."},{"issue":"3","key":"16","first-page":"289","article-title":"On the equation \ud835\udc4e\ud835\udc65\u207f-\ud835\udc4f\ud835\udc66\u207f=\ud835\udc50","volume":"47","author":"Evertse, Jan-Hendrik","year":"1982","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/BF01393828","article-title":"On the representation of integers by binary cubic forms of positive discriminant","volume":"73","author":"Evertse, J.-H.","year":"1983","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"18","unstructured":"[H] S. Hyrr\u00f6, \u201c\u00dcber die Gleichung \ud835\udc4e\ud835\udc65\u207f-\ud835\udc4f\ud835\udc66\u207f=\ud835\udc67 und das Catalansche Problem\u201d, Ann. Acad. Sci. Fenn. Ser. 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Siegel, \u201cDie Gleichung \ud835\udc4e\ud835\udc65\u207f-\ud835\udc4f\ud835\udc66\u207f=\ud835\udc50\u201d, Math. Ann. 144 [1937], 57\u201368.","DOI":"10.1007\/BF01594162"},{"key":"34","unstructured":"[Ta] V. Tartakovski\u012d, \u201cAufl\u00f6sung der Gleichung \ud835\udc65\u2074-\ud835\udf0c\ud835\udc66\u2074=1\u201d, Izv. Akad. Nauk SSR (6) 20 [1926], 301\u2013324."},{"key":"35","doi-asserted-by":"crossref","unstructured":"[Th] A. Thue, \u201c\u00dcber Ann\u00e4herungenswerte algebraischen Zahlen\u201d, J. Reine Angew. 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M.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/9061963753"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00900-4\/S0025-5718-98-00900-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00900-4\/S0025-5718-98-00900-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:41:56Z","timestamp":1776721316000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00900-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":38,"journal-issue":{"issue":"221","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["S0025-5718-98-00900-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00900-4","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":[[1998]]}}}