{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T10:54:42Z","timestamp":1781780082536,"version":"3.54.5"},"reference-count":6,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2025,12]]},"abstract":"Abstract<\/jats:title>\n \n In this paper, we investigate the Diophantine equation\n \n \n $$\\begin{aligned} (2^k - 1)(3^k - 1) = x^n \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n \n (<\/mml:mo>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n \n 3<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n x<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n and prove that it has no solution in positive integers\n \n \n $$k, x, n > 2$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n ,<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s10998-025-00672-y","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T17:05:00Z","timestamp":1753376700000},"page":"582-587","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The diophantine equation $$(2^{k}-1)(3^{k}-1)=x^{n}$$"],"prefix":"10.1007","volume":"91","author":[{"given":"Bo","family":"He","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"issue":"1\u20132","key":"672_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5486\/PMD.2000.2069","volume":"57","author":"L Szalay","year":"2000","unstructured":"L. Szalay, On the Diophantine equation $$\\left(2^{n}-1\\right)\\left(3^{n}-1\\right)=x^{2}$$. Publicationes Mathematicae Debrecen 57(1\u20132), 1\u20139 (2000)","journal-title":"Publicationes Mathematicae Debrecen"},{"key":"672_CR2","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1023\/A:1010335509489","volume":"40","author":"L Hajdu","year":"2000","unstructured":"L. Hajdu, L. Szalay, On the Diophantine Equations $$(2 ^n-1)(6 ^n-1)= x^2$$ and $$(a^n-1)(a ^{kn}-1)= x^2$$. Period. Math. Hung. 40, 141\u2013145 (2000)","journal-title":"Period. Math. Hung."},{"issue":"2","key":"672_CR3","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1023\/A:1019688312555","volume":"44","author":"JHE Cohn","year":"2002","unstructured":"J.H.E. Cohn, The diophantine equation $$(a^n-1)(b^n-1)= x^2$$. Period. Math. Hung. 44(2), 169\u2013175 (2002)","journal-title":"Period. Math. Hung."},{"issue":"1","key":"672_CR4","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/S0022-314X(02)92794-0","volume":"96","author":"F Luca","year":"2002","unstructured":"F. Luca, P.G. Walsh, The product of like-indexed terms in binary recurrences. J. Number Theory 96(1), 152\u2013173 (2002)","journal-title":"J. Number Theory"},{"issue":"4","key":"672_CR5","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/S0019-3577(07)80059-0","volume":"18","author":"MA Bennett","year":"2007","unstructured":"M.A. Bennett, The Diophantine equation $$(x^k-1)(y^k-1)=(z^k-1)^t$$. Indag. Math. 18(4), 507\u2013525 (2007)","journal-title":"Indag. Math."},{"key":"672_CR6","unstructured":"A.H. Parvardi, Lifting the exponent lemma (LTE). Available at url: https:\/\/artofproblemsolving.com\/community\/c6h393335p2198886"}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-025-00672-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10998-025-00672-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-025-00672-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T10:27:52Z","timestamp":1781778472000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10998-025-00672-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,24]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["672"],"URL":"https:\/\/doi.org\/10.1007\/s10998-025-00672-y","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"value":"0031-5303","type":"print"},{"value":"1588-2829","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,24]]},"assertion":[{"value":"17 March 2025","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2025","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2026","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"This article was originally published under the subscription model but it is now published under an Open Access license.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}