{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:01:02Z","timestamp":1760151662248,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T00:00:00Z","timestamp":1649289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Liaoning Educational Committee","award":["2019LNJC08"],"award-info":[{"award-number":["2019LNJC08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Carlitz solved some Diophantine equations on Fibonacci or Lucas numbers. We extend his results to the sequence of generalized Fibonacci and Lucas numbers. In this paper, we solve the Diophantine equations of the form An1\u22efAnk=Bm1\u22efBmrCt1\u22efCts, where (An), (Bm), and (Ct) are generalized Fibonacci or Lucas numbers. Especially, we also find all solutions of symmetric Diophantine equations Ua1Ua2\u22efUam=Ub1Ub2\u22efUbn, where 1&lt;a1\u2264a2\u2264\u22ef\u2264am, and 1&lt;b1\u2264b2\u2264\u22ef\u2264bn.<\/jats:p>","DOI":"10.3390\/sym14040764","type":"journal-article","created":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T22:23:23Z","timestamp":1649370203000},"page":"764","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Carlitz\u2019s Equations on Generalized Fibonacci Numbers"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2725-7415","authenticated-orcid":false,"given":"Min","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Science and Technology Liaoning, Anshan 114051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Science and Technology Liaoning, Anshan 114051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yining","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Science and Technology Liaoning, Anshan 114051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"546046","DOI":"10.1155\/2014\/546046","article-title":"On the stability of a functional equation associated with the Fibonacci numbers","volume":"2014","author":"Mortici","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"137468","DOI":"10.1155\/2014\/137468","article-title":"A linear functional equation of third order associated with the Fibonacci numbers","volume":"2014","author":"Jung","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Andrica, D., and Bagdasar, O. (2021). On Generalized Lucas Pseudoprimality of Level k. Gac. R. Soc. Mat. Esp., 9.","DOI":"10.3390\/math9080838"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley-Interscience.","DOI":"10.1002\/9781118033067"},{"key":"ref_5","first-page":"63","article-title":"lannis Xenakis\u2034 Nomos Alpha\u2033: The Dialectics of Structure and Materials","volume":"24","author":"DeLio","year":"1980","journal-title":"J. Math. 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Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ad\u00e9dji, K.N., Luca, F., and Togb\u00e9, A. (2022). On the solutions of the Diophantine equation Fn \u00b1 a(10m\u22121)9 = k!. J. Number Theory.","DOI":"10.1007\/s40879-023-00626-z"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2307\/1967798","article-title":"On the numerical factors of the arithmetic forms \u03b1n \u00b1 \u03b2n","volume":"15","author":"Carmichael","year":"1913","journal-title":"Ann. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/764\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:49:33Z","timestamp":1760136573000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/764"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,7]]},"references-count":15,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040764"],"URL":"https:\/\/doi.org\/10.3390\/sym14040764","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,7]]}}}