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A note on the Diophantine equation $ x\sp {2p}-Dy\sp 2=1$


Author: Mao Hua Le
Journal: Proc. Amer. Math. Soc. 107 (1989), 27-34
MSC: Primary 11D25; Secondary 11D41
DOI: https://doi.org/10.1090/S0002-9939-1989-0965245-6
MathSciNet review: 965245
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Abstract: Let $ D$ be a positive integer which is square free, and let $ p$ be a prime. In this note we prove that if $ p = 2$ and $ D > \exp 64$ , then the equation $ {x^{2p}} - D{y^2} = 1$ has at most one positive integer solution $ \left( {x,y} \right)$; if $ p > 2$ and $ D > \exp \exp \exp \exp 10$, then every positive integer solution $ \left( {x,y} \right)$ can be expressed as $ {x^p} + y\sqrt D = \varepsilon _1^m$, where $ m$ is a positive integer with $ 2\nmid m,{\varepsilon _1}$ is the fundamental solution of Pell's equation $ {u^2} - D{v^2} = 1$.


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  • [1] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439-444. MR 0234912 (38:3226)
  • [2] -, The theory of linear forms in logarithms, Transcendence theory: Advances and applications, Academic Press, London, 1977, pp. 1-27. MR 0498417 (58:16543)
  • [3] Z. F. Cao, On the diophantine equation $ {x^{2n}} - D{y^2} = 1$ , Proc. Amer. Math. Soc. 98 (1986), 11-16. MR 848864 (87i:11035)
  • [4] Z. Ke and Q. Sun, On the diophantine equation $ {x^4} - D{y^2} = 1$, Acta Math. Sinica 23 (1980), 922-926. MR 615787 (82i:10018)
  • [5] M. H. Le, A necessary and sufficient condition for the equation $ {x^4} - D{y^2} = 1$ to having positive integer solution, Kexue Tongbao 30 (1985), (1986); Changchun Teachers College Acta, Natur. Sci. Ser. (1984), 34-38.
  • [6] -, On the representation of integers by binary quadratic primitive forms, Acta Changchun Teachers College, Natur. Sci. Ser. (1986), 3-12.
  • [7] R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, Reading, Massachusetts, 1983. MR 746963 (86c:11106)
  • [8] W. Ljunggren, Über die Gleichung $ {x^4} - D{y^2} = 1$, Arch. Math. Naturv. 45 (5) (1942), 61-70. MR 0012619 (7:47l)
  • [9] -, Zur Theorie der Gleichung $ {x^2} + 1 = D{y^4}$, Avh. Norske Vid.-Akad. Oslo I(N.S.) 1 (5) (1942).
  • [10] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method, Math. Ann. 231 (1978), 241-267. MR 0460247 (57:242)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0965245-6
Article copyright: © Copyright 1989 American Mathematical Society

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