{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T09:01:19Z","timestamp":1777366879746,"version":"3.51.4"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,11,12]],"date-time":"2016-11-12T00:00:00Z","timestamp":1478908800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2017,12]]},"DOI":"10.1007\/s10998-016-0170-z","type":"journal-article","created":{"date-parts":[[2016,11,12]],"date-time":"2016-11-12T13:47:36Z","timestamp":1478958456000},"page":"143-149","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On the exponential diophantine equation \n                $$\\left( am^{2}+1\\right) ^{x}+\\left( bm^{2}-1\\right) ^{y}=(cm)^{z}$$\n                \n                    \n                                    \n                        \n                            \n                                \n                                    a\n                                    \n                                        m\n                                        2\n                                    \n                                    +\n                                    1\n                                \n                                x\n                            \n                            +\n                            \n                                \n                                    b\n                                    \n                                        m\n                                        2\n                                    \n                                    -\n                                    1\n                                \n                                y\n                            \n                            =\n                            \n                                \n                                    (\n                                    c\n                                    m\n                                    )\n                                \n                                z\n                            \n                        \n                    \n                \n             with \n                $$ c\\mid m $$\n                \n                    \n                                    \n                        \n                            c\n                            \u2223\n                            m"],"prefix":"10.1007","volume":"75","author":[{"given":"Ruiqin","family":"Fu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hai","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,11,12]]},"reference":[{"key":"170_CR1","unstructured":"Y. Bilu, G. Hanrot and P. M. Voutier (with an appendix by M. Mignotte), Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539, 75\u2013122 (2001)"},{"key":"170_CR2","first-page":"55","volume":"539","author":"Y Bugeaud","year":"2001","unstructured":"Y. Bugeaud, T.N. Shorey, On the number of solutions of the generalized Ramanujan\u2013Nagell equation. J. Reine Angew. Math. 539, 55\u201374 (2001)","journal-title":"J. Reine Angew. Math."},{"issue":"4","key":"170_CR3","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1112\/jlms\/s1-39.1.537","volume":"39","author":"JHE Cohn","year":"1964","unstructured":"J.H.E. Cohn, On square Fibonacci numbers. J. Lond. Math. Soc. 39(4), 537\u2013540 (1964)","journal-title":"J. Lond. Math. Soc."},{"key":"170_CR4","volume-title":"Introduction to Number Theory","author":"LK Hua","year":"1982","unstructured":"L.K. Hua, Introduction to Number Theory (Springer, Berlin, 1982)"},{"issue":"2","key":"170_CR5","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1006\/jnth.1995.1138","volume":"55","author":"MH Le","year":"1995","unstructured":"M.H. Le, Some exponential diophantine equations I: the equation \n                        $$D_1x^2-D_2y^2=\\lambda k^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        D\n                                        1\n                                    \n                                    \n                                        x\n                                        2\n                                    \n                                    -\n                                    \n                                        D\n                                        2\n                                    \n                                    \n                                        y\n                                        2\n                                    \n                                    =\n                                    \u03bb\n                                    \n                                        k\n                                        z\n                                    \n                                \n                            \n                        \n                    . J. Number. Theory 55(2), 209\u2013221 (1995)","journal-title":"J. Number. Theory"},{"issue":"3\u20134","key":"170_CR6","first-page":"395","volume":"85","author":"MH Le","year":"2014","unstructured":"M.H. Le, A. Togb\u00e9, H.L. Zhu, On a pure ternary exponential diophantine equation. Publ. Math. Debr. 85(3\u20134), 395\u2013411 (2014)","journal-title":"Publ. Math. Debr."},{"issue":"4","key":"170_CR7","doi-asserted-by":"crossref","first-page":"373","DOI":"10.4064\/aa153-4-3","volume":"153","author":"F Luca","year":"2012","unstructured":"F. Luca, On the system of diophantine equation \n                        $$a^2+b^2=(m^2+1)^r$$\n                        \n                            \n                                            \n                                \n                                    \n                                        a\n                                        2\n                                    \n                                    +\n                                    \n                                        b\n                                        2\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        r\n                                    \n                                \n                            \n                        \n                     and \n                        $$a^x+b^y=(m^2+1)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        a\n                                        x\n                                    \n                                    +\n                                    \n                                        b\n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    , Acta Arith. 153(4), 373\u2013392 (2012)","journal-title":"Acta Arith."},{"issue":"2","key":"170_CR8","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1016\/j.jnt.2012.08.018","volume":"133","author":"T Miyazaki","year":"2013","unstructured":"T. Miyazaki, Generalizations of classical results on Je\u015bmanowicz\u2019 conjecture concerning Pythagorean triples. J. Number Theory 133(2), 583\u2013589 (2013)","journal-title":"J. Number Theory"},{"issue":"1","key":"170_CR9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.4064\/aa164-1-3","volume":"164","author":"T Miyazaki","year":"2014","unstructured":"T. Miyazaki, A note on the article by F. Luca \u201cOn the system of diophantine equations \n                        $$a^2+b^2=(m^2+1)^r$$\n                        \n                            \n                                            \n                                \n                                    \n                                        a\n                                        2\n                                    \n                                    +\n                                    \n                                        b\n                                        2\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        r\n                                    \n                                \n                            \n                        \n                     and \n                        $$a^x+b^y=(m^2+1)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        a\n                                        x\n                                    \n                                    +\n                                    \n                                        b\n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    \u201d. Acta Arith. 164(1), 31\u201342 (2014)","journal-title":"Acta Arith."},{"issue":"1","key":"170_CR10","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1017\/S0004972713000956","volume":"90","author":"T Miyazaki","year":"2014","unstructured":"T. Miyazaki, N. Terai, on the exponential diophantine equation \n                        $$(m^2+1)^x+(cm^2-1)^y=(am)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        \n                                            (\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        x\n                                    \n                                    +\n                                    \n                                        \n                                            (\n                                            c\n                                            \n                                                m\n                                                2\n                                            \n                                            -\n                                            1\n                                            )\n                                        \n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            a\n                                            m\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    , Bull. Aust. Math. Soc. 90(1), 9\u201319 (2014)","journal-title":"Bull. Aust. Math. Soc."},{"issue":"8","key":"170_CR11","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.1142\/S1793042112501151","volume":"8","author":"T Miyazaki","year":"2012","unstructured":"T. Miyazaki, A. Togb\u00e9, The diophantine equation \n                        $$(2am-1)^x+(2m)^y=(2am+1)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        \n                                            (\n                                            2\n                                            a\n                                            m\n                                            -\n                                            1\n                                            )\n                                        \n                                        x\n                                    \n                                    +\n                                    \n                                        \n                                            (\n                                            2\n                                            m\n                                            )\n                                        \n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            2\n                                            a\n                                            m\n                                            +\n                                            1\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    , Int. J. Number Theory 8(8), 2035\u20132044 (2012)","journal-title":"Int. J. Number Theory"},{"issue":"1","key":"170_CR12","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jnt.2014.01.011","volume":"141","author":"T Miyazaki","year":"2014","unstructured":"T. Miyazaki, P.Z. Yuan, D.Y. Wu, Generalizations of classical results on Je\u015bmanowicz\u2019 conjecture concerning Pythagorean triples II, J. Number Theory 141(1), 184\u2013201 (2014)","journal-title":"J. Number Theory"},{"issue":"21\u201324","key":"170_CR13","first-page":"1135","volume":"6","author":"N Terai","year":"2012","unstructured":"N. Terai, On the exponential diophantine equation \n                        $$(4m^2+1)^x+(5m^2-1)^y=(3m)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        \n                                            (\n                                            4\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        x\n                                    \n                                    +\n                                    \n                                        \n                                            (\n                                            5\n                                            \n                                                m\n                                                2\n                                            \n                                            -\n                                            1\n                                            )\n                                        \n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            3\n                                            m\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    , Int. J. Algbra 6(21\u201324), 1135\u20131146 (2012)","journal-title":"Int. J. Algbra"},{"issue":"2","key":"170_CR14","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.jnt.2014.02.009","volume":"141","author":"N Terai","year":"2014","unstructured":"N. Terai, On Je\u015bmanowicz\u2019 conjecture concerning Pythagorean triples. J. Number Theory 141(2), 316\u2013323 (2014)","journal-title":"J. Number Theory"},{"issue":"1","key":"170_CR15","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.jnt.2015.04.009","volume":"156","author":"H Yang","year":"2015","unstructured":"H. Yang, R.Q. Fu, A note on Je\u015bmanowicz\u2019 conjecture concerning primitive Pythagorean triples. J. Number Theory 156(1), 183\u2013194 (2015)","journal-title":"J. Number Theory"},{"key":"170_CR16","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1090\/S0025-5718-1995-1284673-6","volume":"64","author":"PM Voutier","year":"1995","unstructured":"P.M. Voutier, Primitive divisors of Lucas and Lehmer sequences. Math. Comp. 64, 869\u2013888 (1995)","journal-title":"Math. Comp."},{"issue":"1","key":"170_CR17","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4064\/cm139-1-7","volume":"139","author":"JP Wang","year":"2015","unstructured":"J.P. Wang, T.T. Wang, W.P. Zhang, A note on the exponential diophantine equation \n                        $$(4m^2+1)^x+(5m^2-1)^y=(3m)^z$$\n                        \n                            \n                                            \n                                \n                                    \n                                        \n                                            (\n                                            4\n                                            \n                                                m\n                                                2\n                                            \n                                            +\n                                            1\n                                            )\n                                        \n                                        x\n                                    \n                                    +\n                                    \n                                        \n                                            (\n                                            5\n                                            \n                                                m\n                                                2\n                                            \n                                            -\n                                            1\n                                            )\n                                        \n                                        y\n                                    \n                                    =\n                                    \n                                        \n                                            (\n                                            3\n                                            m\n                                            )\n                                        \n                                        z\n                                    \n                                \n                            \n                        \n                    , Colloq. Math. 139(1), 121\u2013126 (2015)","journal-title":"Colloq. Math."}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10998-016-0170-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-016-0170-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-016-0170-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,10,31]],"date-time":"2017-10-31T09:29:27Z","timestamp":1509442167000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10998-016-0170-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,12]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,12]]}},"alternative-id":["170"],"URL":"https:\/\/doi.org\/10.1007\/s10998-016-0170-z","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"value":"0031-5303","type":"print"},{"value":"1588-2829","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11,12]]}}}