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%I #43 May 07 2021 00:41:51
%S 2,5234,8158,93844,367806,421351,720114,939787,28187351,110781386,
%T 154319269,384242766,390620082,3790689201,65589428378,952764389446,
%U 12438517260105,35495694227489,53197086958290,5853886516781223
%N Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.
%C Hall conjectured that the nonzero difference k = x^3 - y^2 cannot be less than C x^(1/2), for a constant C. His original conjecture, probably false, has been reformulated in the following way: For any exponent e < 1/2, a constant K_e > 0 exists such that |x^3 - y^2| > K_e x^e.
%C Danilov found an infinite family of solutions to |x^3 - y^2| < sqrt(x). For more detail see A200216. [_Artur Jasinski_, Nov 04 2011]
%D Noam D. Elkies, Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction. Algorithmic Number Theory. Proceedings of ANTS-IV; W. Bosma, ed.; Springer, 2000; pp. 33-63.
%D Marshall Hall Jr., The Diophantine equation x^3 - y^2 = k, in Computers in Number Theory; A. O. L. Atkin and B. Birch, eds.; Academic Press, 1971; pp. 173-198.
%H Ismael Jimenez Calvo, <a href="https://web.archive.org/web/20191127011622/http://ijcalvo.galeon.com/hall.htm">Marshall Hall's conjecture</a>.
%H Ismael Jimenez Calvo and G. Saez Moreno, <a href="http://dx.doi.org/10.1007/3-540-45439-X_21">Approximate Power roots in Z_m</a>, Proceedings of ISC 2001 (Information Security); G. I. Davida and Y. Frankel, eds.; Springer, 2001; pp. 310-323.
%H I. Jiminez Calvo, J. Herranz, and G. Saez, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02240-6">A new algorithm to search for small nonzero |x^3-y^2| values</a>, Math. Comp. 76 (268) (2009) 2435-2444.
%H L. V. Danilov, <a href="http://dx.doi.org/10.1007/BF01140190">Diophantine equation x^3-y^2-k and Hall's conjecture</a>, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
%H L. V. Danilov, <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=5944&option_lang=eng">Letter to the editors</a>, Mat. Zametki, 36:3 (1984), 457-458.
%H L. V. Danilov, <a href="http://dx.doi.org/10.1007/BF01141949">Letter to the editor</a>, Mathem. Notes, 36 (3) (1984), 726.
%H R. D'Mello, <a href="http://arxiv.org/abs/1410.0078">Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves</a>, arXiv preprint arXiv:1410.0078 [math.NT], 2014.
%H Noam D. Elkies, <a href="http://www.math.harvard.edu/~elkies/hall.html">List of integers x,y with x<10^18, 0 < |x^3-y^2| < x^(1/2).</a>
%H J. Gebel, A. Petho and H. G. Zimmer, <a href="http://dx.doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Math. 110 (1998), 335-367.
%e |5234^3 - 378661^2| = 17 < sqrt(5234), so 5234 is in the sequence.
%t For[x=1, True, x++, If[Abs[x^3-Round[Sqrt[x^3]]^2] < Sqrt[x] && !IntegerQ[Sqrt[x]], Print[x]]]
%Y Cf. A179108, A179387, A200216.
%K nonn,more
%O 1,1
%A _Dean Hickerson_ and _Robert G. Wilson v_, Dec 16 2002
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