{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:10:49Z","timestamp":1776802249104,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"305","license":[{"start":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T00:00:00Z","timestamp":1503014400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-14-CE25-0015"],"award-info":[{"award-number":["ANR-14-CE25-0015"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -units, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables us to carry out such a program explicitly, at least for certain small sets of primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; we do so for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S equals StartSet 2 comma 3 EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S = \\{ 2, 3 \\}<\/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=\"upper S equals StartSet 3 comma 5 comma 7 EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>7<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S= \\{ 3, 5, 7 \\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3129","type":"journal-article","created":{"date-parts":[[2016,7,14]],"date-time":"2016-07-14T12:08:22Z","timestamp":1468498102000},"page":"1375-1401","source":"Crossref","is-referenced-by-count":3,"title":["Sums of two \ud835\udc46-units via Frey-Hellegouarch curves"],"prefix":"10.1090","volume":"86","author":[{"given":"Michael","family":"Bennett","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicolas","family":"Billerey","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,8,18]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1112\/S0024609304003480","article-title":"Products of consecutive integers","volume":"36","author":"Bennett, Michael A.","year":"2004","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1112\/S0010437X06002181","article-title":"Binomial Thue equations and polynomial powers","volume":"142","author":"Bennett, M. A.","year":"2006","journal-title":"Compos. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"23","DOI":"10.4153\/CJM-2004-002-2","article-title":"Ternary Diophantine equations via Galois representations and modular forms","volume":"56","author":"Bennett, Michael A.","year":"2004","journal-title":"Canad. J. 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