{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:53:18Z","timestamp":1776837198747,"version":"3.51.2"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"344","license":[{"start":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T00:00:00Z","timestamp":1715731200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I4406"],"award-info":[{"award-number":["I4406"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I4406"],"award-info":[{"award-number":["I4406"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I4406"],"award-info":[{"award-number":["I4406"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I4406"],"award-info":[{"award-number":["I4406"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis upper U Subscript n Baseline right-parenthesis Subscript n element-of double-struck upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>U<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(U_n)_{n\\in \\mathbb {N}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/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 N 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">N_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that for any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b comma c element-of double-struck upper Z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b,c\\in \\mathbb {Z}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b greater-than upper B\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b&gt; B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper U Subscript n Baseline minus b Superscript m Baseline equals c\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>U<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">U_n - b^m = c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has at most two distinct solutions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n comma m right-parenthesis element-of double-struck upper N squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n,m)\\in \\mathbb {N}^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to upper N 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq N_0<\/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=\"m greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m\\geq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Moreover, we apply our result to the special case of Tribonacci numbers given by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1 equals upper T 2 equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1= T_2=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 3 equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_3=2<\/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 T Subscript n Baseline equals upper T Subscript n minus 1 Baseline plus upper T Subscript n minus 2 Baseline plus upper T Subscript n minus 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_{n}=T_{n-1}+T_{n-2}+T_{n-3}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . By means of the LLL-algorithm and continued fraction reduction we are able to prove\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N 0 equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N_0=2<\/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 B equals e Superscript 438\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>e<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>438<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B=e^{438}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The corresponding reduction algorithm is implemented in Sage.\n                  <\/p>","DOI":"10.1090\/mcom\/3854","type":"journal-article","created":{"date-parts":[[2023,4,12]],"date-time":"2023-04-12T10:34:35Z","timestamp":1681295675000},"page":"2825-2859","source":"Crossref","is-referenced-by-count":2,"title":["On the Diophantine equation \ud835\udc48_{\ud835\udc5b}-\ud835\udc4f^{\ud835\udc5a}=\ud835\udc50"],"prefix":"10.1090","volume":"92","author":[{"given":"Sebastian","family":"Heintze","sequence":"first","affiliation":[]},{"given":"Robert","family":"Tichy","sequence":"additional","affiliation":[]},{"given":"Ingrid","family":"Vukusic","sequence":"additional","affiliation":[]},{"given":"Volker","family":"Ziegler","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,5,15]]},"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":"2","key":"2","doi-asserted-by":"publisher","first-page":"185","DOI":"10.3336\/gm.57.2.02","article-title":"On the multiplicity in Pillai\u2019s problem with Fibonacci numbers and powers of a fixed prime","volume":"57(77)","author":"Batte, Herbert","year":"2022","journal-title":"Glas. Mat. Ser. III","ISSN":"https:\/\/id.crossref.org\/issn\/0017-095X","issn-type":"print"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"897","DOI":"10.4153\/CJM-2001-036-6","article-title":"On some exponential equations of S. S. Pillai","volume":"53","author":"Bennett, Michael A.","year":"2001","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"issue":"10","key":"4","doi-asserted-by":"publisher","first-page":"1434","DOI":"10.1080\/10236198.2021.1990900","article-title":"Pillai\u2019s problem with \ud835\udc58-Fibonacci and Pell numbers","volume":"27","author":"Bravo, Jhon J.","year":"2021","journal-title":"J. Difference Equ. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1023-6198","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.4134\/BKMS.b160486","article-title":"On Pillai\u2019s problem with Tribonacci numbers and powers of 2","volume":"54","author":"Bravo, Jhon J.","year":"2017","journal-title":"Bull. Korean Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/1015-8634","issn-type":"print"},{"issue":"7","key":"6","doi-asserted-by":"publisher","first-page":"1711","DOI":"10.1142\/S1793042117500981","article-title":"On a variant of Pillai\u2019s problem","volume":"13","author":"Chim, Kwok Chi","year":"2017","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.jnt.2017.07.016","article-title":"On a variant of Pillai\u2019s problem II","volume":"183","author":"Chim, Kwok Chi","year":"2018","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1556\/012.2019.56.3.1435","article-title":"On the problem of Pillai with Padovan numbers and powers of 3","volume":"56","author":"Ddamulira, Mahadi","year":"2019","journal-title":"Studia Sci. Math. Hungar.","ISSN":"https:\/\/id.crossref.org\/issn\/0081-6906","issn-type":"print"},{"issue":"5","key":"9","first-page":"Art. 19.5.6, 14","article-title":"On the problem of Pillai with tribonacci numbers and powers of 3","volume":"22","author":"Ddamulira, Mahadi","year":"2019","journal-title":"J. Integer Seq."},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s40590-019-00263-1","article-title":"On a problem of Pillai with Fibonacci numbers and powers of 3","volume":"26","author":"Ddamulira, Mahadi","year":"2020","journal-title":"Bol. Soc. Mat. Mex. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/1405-213X","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s00605-018-1155-1","article-title":"On a problem of Pillai with \ud835\udc58-generalized Fibonacci numbers and powers of 2","volume":"187","author":"Ddamulira, Mahadi","year":"2018","journal-title":"Monatsh. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-9255","issn-type":"print"},{"issue":"7","key":"12","doi-asserted-by":"publisher","first-page":"1643","DOI":"10.1142\/S1793042120500876","article-title":"On the problem of Pillai with \ud835\udc58-generalized Fibonacci numbers and powers of 3","volume":"16","author":"Ddamulira, Mahadi","year":"2020","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s12044-017-0338-3","article-title":"On a problem of Pillai with Fibonacci numbers and powers of 2","volume":"127","author":"Ddamulira, Mahadi","year":"2017","journal-title":"Proc. Indian Acad. Sci. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0253-4142","issn-type":"print"},{"issue":"10","key":"14","doi-asserted-by":"publisher","first-page":"2251","DOI":"10.1142\/S1793042121500871","article-title":"On Pillai\u2019s problem with \ud835\udc4b-coordinates of Pell equations and powers of 2 II","volume":"17","author":"Erazo, Harold S.","year":"2021","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s40590-018-0223-9","article-title":"On Pillai\u2019s problem with the Fibonacci and Pell sequences","volume":"25","author":"Hern\u00e1ndez, Santos Hern\u00e1ndez","year":"2019","journal-title":"Bol. Soc. Mat. Mex. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/1405-213X","issn-type":"print"},{"key":"16","first-page":"22","article-title":"On Pillai\u2019s problem with Pell numbers and powers of 2","volume":"41","author":"Hernane, Mohand Ouamar","year":"2018","journal-title":"Hardy-Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/2804-7370","issn-type":"print"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"325","DOI":"10.4064\/aa133-4-3","article-title":"Linear forms in two logarithms and interpolation determinants. 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