Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
MathSciNet review: 1152282
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [1] J. Blass, On the Diophantine equation 𝑌²+𝐾=𝑋⁵, Bull. Amer. Math. Soc. 80 (1974), 329. MR 0330041, 10.1090/S0002-9904-1974-13487-7
  • [2] Josef Blass and Ray Steiner, On the equation 𝑦²+𝑘=𝑥⁷, Utilitas Math. 13 (1978), 293–297. MR 0480327
  • [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] 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
  • [9] Wilhelm Ljunggren, On the Diophantine equation 𝑥²+𝐷=𝑦ⁿ, Norske Vid. Selsk. Forh., Trondhjem 17 (1944), no. 23, 93–96. MR 0019644
  • [10] Wilhelm Ljunggren, On a Diophantine equation, Norske Vid. Selsk. Forh., Trondhjem 18 (1945), no. 32, 125–128. MR 0017304
  • [11] Wilhelm Ljunggren, New theorems concerning the diophantine equation 𝐶𝑥²+𝐷=𝑦ⁿ, Norske Vid. Selsk. Forh., Trondheim 29 (1956), 1–4. MR 0078388
  • [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] W. Ljunggren, New theorems concerning the diophantine equation 𝑥²+𝐷=4𝑦^{𝑞}, Acta Arith. 21 (1972), 183–191. MR 0302557
  • [15] Maurice Mignotte and Michel Waldschmidt, Linear forms in two logarithms and Schneider’s method. III, Ann. Fac. Sci. Toulouse Math. (5) suppl. (1989), 43–75 (English, with English and French summaries). MR 1425750
  • [16] T. Nagell, Sur l'impossibilité de quelques équation á deux indéterminées, Norsk Mat. Forenings Skrifter Ser. 1 13 (1923), 65-82.
  • [17] Trygve Nagell, 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), no. 2, 38. MR 0070645
  • [18] Bengt Persson, On a Diophantine equation in two unknowns, Ark. Mat. 1 (1949), 45–57. MR 0032670
  • [19] Bengt Stolt, Die Anzahl von Lösungen gewisser diophantischer Gleichungen, Arch. Math. 8 (1957), 393–400 (German). MR 0103169

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11D61

Retrieve articles in all journals with MSC: 11D61


Additional Information

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