Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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A note on Mordell’s equation $y^{2}=x^{3}+k$
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by S. P. Mohanty
Proc. Amer. Math. Soc. 39 (1973), 645-646
DOI: https://doi.org/10.1090/S0002-9939-1973-0316377-5

Abstract:

In this note we prove that $\lim {\sup _{k \to \infty }}N’(k) \geqq 6$, where $N’(k)$ is the number of integral solutions of ${y^2} = {x^3} + k$ with $(x,y) = 1$.
References
  • L. J. Mordell, On some Diophantine equations $y^2=x^3+k$ with no rational solutions, Arch. Math. Naturvid. 49 (1947), no. 6, 143–150. MR 22857
  • L. J. Mordell, The infinity of rational solutions of $y^{2}=x^{3}+k$, J. London Math. Soc. 41 (1966), 523–525. MR 197394, DOI 10.1112/jlms/s1-41.1.523
  • J. P. Serre, $P$-torsion des courbes elliptiques, Seminaire Bourbaki no. 380 (1970).
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Bibliographic Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 39 (1973), 645-646
  • MSC: Primary 10B10; Secondary 14G25, 14H25
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0316377-5
  • MathSciNet review: 0316377