Abstract
For anyD 1,D 2∈ℕ, leth(-D 1 D 2) denote the class number of the imaginary quadratic field\(\mathbb{Q}(\sqrt { - D_1 D_2 } )\). In this paper we prove that the equationD 1 x 2+D m2 =4y n.D 1,D 2,x, y, m, n∈, gcd (D 1x,D 2y=1,2Χm,n an odd prime,nΧh(-D 1 D 2, has only a finite number of solutions (D 1,D 2,x,y,m,n) withn>5. Moreover, the solutions satisfy 4y n<exp exp 470.
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Brtindza, B., Van der Poorten, A. J., Waldschmidt, M.: On the distribution and number of solutions of Thue's equation. In preparation.
Györy, K., Papp, Z. Z.: Norm form equations and explicit lower bounds for linear forms with algebraic coefficients. In: (To the memory of Paul Turán) Studies in Pure Mathematics, pp. 245–247. Budapest: Akadémiai Kiadó. 1983.
Le, M.-H.: A diophantine equation concerning the divisibility of the class number for some imaginary quadratic fields. Indag. Math. (N.S.)4, 65–70 (1993).
Lidl, R., Niederreiter, H.: Finite Fields. Reading, MA: Addison-Wesley. 1983.
Lijunggren, W.: On the diophantine equationx 2+D=4y n. Mh. Math.75, 136–143 (1971).
Ljunggren, W.: New theorems concerning the diophantine equationx 2+D=4y n. Acta Arith.2, 183–191 (1972).
Mignotte, M., Waldschmidt, M.: Linear forms in two logarithms and Schneider's method (III). Ann. Fac. Sci. Toulouse97, 43–75 (1990).
Persson, B.: On a diophantine equation in two unknowns. Ark. Mat.1, 45–57 (1949).
Stolt, B.: Die Anzahl von Lösungen gewisser diophantischer Gleichungen. Arch. Math.8, 393–400 (1957_.
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Supported by the National Natural Science Foundation of China
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Maohua, L. On the diophantine equationD 1 x 2+D m2 =4y n . Monatshefte für Mathematik 120, 121–125 (1995). https://doi.org/10.1007/BF01585912
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DOI: https://doi.org/10.1007/BF01585912