{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:44:21Z","timestamp":1776725061206,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"230","license":[{"start":{"date-parts":[[2000,5,24]],"date-time":"2000-05-24T00:00:00Z","timestamp":959126400000},"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":"

\n For an integral parameter\n \n \n \n \n t<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n t \\in \\mathbb {Z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n we investigate the family of Thue equations\n \n \n \n \n \n \n F<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n x<\/mml:mi>\n \n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n (<\/mml:mo>\n t<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n )<\/mml:mo>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n x<\/mml:mi>\n \n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n y<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n (<\/mml:mo>\n 2<\/mml:mn>\n \n t<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 4<\/mml:mn>\n t<\/mml:mi>\n +<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n \n x<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n y<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \u00a0<\/mml:mtext>\n +<\/mml:mo>\n (<\/mml:mo>\n \n t<\/mml:mi>\n \n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n t<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 2<\/mml:mn>\n \n t<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 4<\/mml:mn>\n t<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 3<\/mml:mn>\n )<\/mml:mo>\n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n y<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n (<\/mml:mo>\n \n t<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n t<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 5<\/mml:mn>\n t<\/mml:mi>\n +<\/mml:mo>\n 3<\/mml:mn>\n )<\/mml:mo>\n x<\/mml:mi>\n \n y<\/mml:mi>\n \n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n y<\/mml:mi>\n \n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n =<\/mml:mo>\n \n \u00b1\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n ,<\/mml:mo>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n \\begin{multline*} F(x,y) = x^{5} + (t-1)^{2}x^{4}y - (2t^{3}+4t+4)x^{3}y^{2}\\ + (t^{4}+t^{3}+2t^{2}+4t-3)x^{2}y^{3} + (t^{3}+t^{2}+5t+3)xy^{4} + y^{5} = \\pm 1\\,, \\end{multline*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n originating from Emma Lehmer\u2019s family of quintic fields, and show that for\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n t<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2265\n \n <\/mml:mo>\n 3.28<\/mml:mn>\n \n \u22c5\n \n <\/mml:mo>\n \n 10<\/mml:mn>\n \n 15<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n |t| \\ge 3.28 \\cdot 10^{15}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the only solutions are the trivial ones with\n \n \n \n \n x<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n x=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n y<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n y=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.\n <\/p>","DOI":"10.1090\/s0025-5718-99-01155-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"851-859","source":"Crossref","is-referenced-by-count":3,"title":["A parametric family of quintic Thue equations"],"prefix":"10.1090","volume":"69","author":[{"given":"Istv\u00e1n","family":"Ga\u00e1l","sequence":"first","affiliation":[]},{"given":"G\u00fcnter","family":"Lettl","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1999,5,24]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1515\/crll.1993.442.19","article-title":"Logarithmic forms and group varieties","volume":"442","author":"Baker, A.","year":"1993","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"194","key":"2","doi-asserted-by":"publisher","first-page":"795","DOI":"10.2307\/2008408","article-title":"Note on a polynomial of Emma Lehmer","volume":"56","author":"Darmon, Henri","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"220","key":"3","doi-asserted-by":"publisher","first-page":"1689","DOI":"10.1090\/S0025-5718-97-00868-5","article-title":"Power integral bases in a parametric family of totally real cyclic quintics","volume":"66","author":"Ga\u00e1l, Istv\u00e1n","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"C. Heuberger, On a family of quintic Thue equations, J. Symbolic Comput. 26 (1998), 173\u2013185.","DOI":"10.1006\/jsco.1998.0206"},{"key":"5","unstructured":"C. Heuberger, A. Peth\u0151 & R.F. Tichy, Complete solution of parametrized Thue equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), 93\u2013114."},{"issue":"182","key":"6","doi-asserted-by":"publisher","first-page":"535","DOI":"10.2307\/2008622","article-title":"Connection between Gaussian periods and cyclic units","volume":"50","author":"Lehmer, Emma","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/BF02953340","article-title":"Complete solution of a family of quartic Thue equations","volume":"65","author":"Lettl, G.","year":"1995","journal-title":"Abh. Math. Sem. Univ. Hamburg","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5858","issn-type":"print"},{"issue":"213","key":"8","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1090\/S0025-5718-96-00662-X","article-title":"Complete solutions of a family of quartic Thue and index form equations","volume":"65","author":"Mignotte, Maurice","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"196","key":"9","doi-asserted-by":"publisher","first-page":"777","DOI":"10.2307\/2938717","article-title":"Complete solutions to families of quartic Thue equations","volume":"57","author":"Peth\u0151, Attila","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"A. Peth\u0151 & R.F. Tichy, On two-parametric quartic families of Diophantine problems, J. Symbolic Comput. 26 (1998), 151\u2013171.","DOI":"10.1006\/jsco.1998.0205"},{"issue":"182","key":"11","doi-asserted-by":"publisher","first-page":"535","DOI":"10.2307\/2008622","article-title":"Connection between Gaussian periods and cyclic units","volume":"50","author":"Lehmer, Emma","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/0022-314X(90)90154-J","article-title":"Complete solutions to a family of cubic Diophantine equations","volume":"34","author":"Thomas, Emery","year":"1990","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1006\/jnth.1993.1024","article-title":"Solutions to certain families of Thue equations","volume":"43","author":"Thomas, Emery","year":"1993","journal-title":"J. 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