A note on the Diophantine equation $x^ {2p}-Dy^ 2=1$
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- by Mao Hua Le PDF
- Proc. Amer. Math. Soc. 107 (1989), 27-34 Request permission
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 ${u^2} - D{v^2} = 1$.References
- A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439–444. MR 234912, DOI 10.1017/s0305004100044418
- A. Baker, The theory of linear forms in logarithms, Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976) Academic Press, London, 1977, pp. 1–27. MR 0498417
- Zhen Fu Cao, On the Diophantine equation $x^{2n}-{\scr D}y^2=1$, Proc. Amer. Math. Soc. 98 (1986), no. 1, 11–16. MR 848864, DOI 10.1090/S0002-9939-1986-0848864-4
- Zhao Ke and Qi Sun, On the Diophantine equation $x^{4}-Dy^{2}=1$, Acta Math. Sinica 23 (1980), no. 6, 922–926 (Chinese). MR 615787 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. —, On the representation of integers by binary quadratic primitive forms, Acta Changchun Teachers College, Natur. Sci. Ser. (1986), 3-12.
- Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
- Wilhelm Ljunggren, Über die Gleichung $x^4-Dy^2=1$, Arch. Math. Naturvid. 45 (1942), no. 5, 61–70 (German). MR 12619 —, Zur Theorie der Gleichung ${x^2} + 1 = D{y^4}$, Avh. Norske Vid.-Akad. Oslo I(N.S.) 1 (5) (1942).
- Maurice Mignotte and Michel Waldschmidt, Linear forms in two logarithms and Schneider’s method, Math. Ann. 231 (1977/78), no. 3, 241–267. MR 460247, DOI 10.1007/BF01420244
Additional Information
- © Copyright 1989 American Mathematical Society
- 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