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In this paper, we look at the exponential Diophantine equation <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$L_n-2^x3^y=c$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>x<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:msup>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mi>y<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>c<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, for <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n,x,y\\in \\mathbb {Z}_{\\ge 0}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>Z<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>\u2265<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. We treat the cases <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$c\\in -\\mathbb {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$c=0$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$c\\in \\mathbb {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> independently. In the cases that <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$c\\in \\mathbb {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$c\\in -\\mathbb {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, we find all integers <jats:italic>c<\/jats:italic> such that the Diophantine equation has at least three solutions. These cases are treated independently, since we employ quite different techniques in proving the two cases.<\/jats:p>","DOI":"10.1007\/s10998-025-00649-x","type":"journal-article","created":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T08:40:07Z","timestamp":1748680807000},"page":"53-87","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On a problem of Pillai involving S-units and Lucas numbers"],"prefix":"10.1007","volume":"91","author":[{"given":"Herbert","family":"Batte","sequence":"first","affiliation":[]},{"given":"Mahadi","family":"Ddamulira","sequence":"additional","affiliation":[]},{"given":"Juma","family":"Kasozi","sequence":"additional","affiliation":[]},{"given":"Florian","family":"Luca","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,31]]},"reference":[{"key":"649_CR1","first-page":"19","volume":"442","author":"A Baker","year":"1993","unstructured":"A. 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