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On the Diophantine equations $ d\sb 1x\sp 2+2\sp {2m}d\sb 2=y\sp n$ and $ d\sb 1x\sp 2+d\sb 2=4y\sp n$


Author: Le Maohua
Journal: Proc. Amer. Math. Soc. 118 (1993), 67-70
MSC: Primary 11D61
DOI: https://doi.org/10.1090/S0002-9939-1993-1152282-5
MathSciNet review: 1152282
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Abstract: Let $ {d_1},\;{d_2}$ be coprime positive integers, which are squarefree, and let $ h$ denote the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt { - {d_1}{d_2}} )$. Let $ m,\;n$ be integers such that $ m \geqslant 0,\;n > 1$, and $ \gcd (n,2h) = 1$. In this paper we prove that if $ n \geqslant 8.5 \cdot {10^6}$, then the equations $ {d_1}{x^2} + {2^{2m}}{d_2} = {y^n}(2\nmid y)$ and $ {d_1}{x^2} + {d_2} = 4{y^n}$ have no positive integer solutions $ (x,y)$ with $ \gcd (x,y) = 1$.


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  • [1] J. Blass, On the diophantine equation $ {Y^2} + K = {X^5}$, Bull. Amer. Math. Soc. 80 (1974), 329. MR 0330041 (48:8380)
  • [2] J. Blass and R. Steiner, On the equation $ {y^2} + k = {x^7}$, Utilitas Math. 13 (1978), 293-297. MR 0480327 (58:500)
  • [3] L. Cardell, Some results on the diophantine equation $ {x^2} + D = {y^n}$, Dep. Math., Chalmers Univ., Techn. Univ. Göteborg 1984-06, 1984.
  • [4] O. Korhonen, On the diophantine equation $ A{x^2} + 8B = {y^n}$, Acta Univ. Oulu Ser. A Sci. Rerum. Natur. Math. 16 (1979).
  • [5] -, On the diophantine equation $ A{x^2} + 2B = {y^n}$, Acta Univ. Oulu Ser. A Sci. Rerum. Natur. Math. 17 (1979).
  • [6] -, On the diophantine equation $ 2A{x^2} + B = {y^n}$, Acta Univ. Oulu Ser. A Sci. Rerum. Natur. Math. 21 (1980).
  • [7] -, On the diophantine equation $ C{x^2} + D = {y^n}$, Acta Univ. Oulu Ser. A Sci. Rerum. Natur. Math. 25 (1981).
  • [8] R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, Reading, MA, 1983. MR 746963 (86c:11106)
  • [9] W. Ljunggren, On the diophantine equation $ {x^2} + D = {y^n}$, Norske Vid. Selsk. Forh. Trondheim 17 (1944), 37-43. MR 0019644 (8:442e)
  • [10] -, On a diophantine equation, Norske Vid. Selsk. Forh. Trondheim 18 (1945), 125-128. MR 0017304 (8:136a)
  • [11] -, New theorems concerning the diophantine equation $ C{x^2} + D = {y^n}$, Norske Vid. Selsk. Forh. Trondheim 29 (1956), 1-4. MR 0078388 (17:1185g)
  • [12] -, On the diophantine equation $ C{x^2} + D = {y^n}$, Pacific J. Math. 14 (1964), 585-596.
  • [13] -, On the diophantine equation $ {x^2} + D = 4{y^9}$, Monatsh. Math. 75 (1971), 136-143.
  • [14] -, New theorems concerning the diophantine equation $ {x^2} + D = 4{y^n}$, Acta Arith. 21 (1972), 183-191. MR 0302557 (46:1701)
  • [15] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method. III, Ann. Fac. Sci. Toulouse (1990), 43-75. MR 1425750 (98i:11055)
  • [16] T. Nagell, Sur l'impossibilité de quelques équation á deux indéterminées, Norsk Mat. Forenings Skrifter Ser. 1 13 (1923), 65-82.
  • [17] -, Contributions to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta Soc. Sci. Upsal. (4) 16 (1955). MR 0070645 (17:13b)
  • [18] B. Persson, On a diophantine equation in two unknowns, Ark. Mat. 1 (1949), 45-57. MR 0032670 (11:328e)
  • [19] B. Stolt, Die Anzahl von Lösungen gewisser diophantischer Gleichungen, Arch. Math. 8 (1957), 393-400. MR 0103169 (21:1952)

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DOI: https://doi.org/10.1090/S0002-9939-1993-1152282-5
Article copyright: © Copyright 1993 American Mathematical Society

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