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A078933
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Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.
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12
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2, 5234, 8158, 93844, 367806, 421351, 720114, 939787, 28187351, 110781386, 154319269, 384242766, 390620082, 3790689201, 65589428378, 952764389446, 12438517260105, 35495694227489, 53197086958290, 5853886516781223
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
Danilov found an infinite family of solutions to |x^3 - y^2| < sqrt(x). For more detail see A200216. [from Artur Jasinski, Nov 04 2011]
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REFERENCES
| 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.
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.
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LINKS
| Ismael Jimenez Calvo, Hall's conjecture.
Ismael Jimenez Calvo and G. Saez Moreno, Approximate Power roots in Z_m, Proceedings of ISC 2001 (Information Security); G. I. Davida and Y. Frankel, eds.; Springer, 2001; pp. 310-323.
I. Jiminez Calvo, J. Herranz, G. Saez, A new algorithm to search for small nonzero |x^3-y^2| values, Math. Comp. 76 (268) (2009) 2435-2444.
L. V. Danilov Diophantine equation x^3-y^2-k and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
L. V. Danilov, Letter to the editors, Mat. Zametki, 36:3 (1984), 457-458.
L. V. Danilov, Letter to the editor, Mathem. Notes, 36 (3) (1984), 726.
Noam D. Elkies, List of integers x,y with x<10^18, 0 < |x^3-y^2| < x^(1/2).
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
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EXAMPLE
| |5234^3 - 378661^2| = 17 < sqrt(5234), so 5234 is in the sequence.
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MATHEMATICA
| For[x=1, True, x++, If[Abs[x^3-Round[Sqrt[x^3]]^2] < Sqrt[x] && !IntegerQ[Sqrt[x]], Print[x]]]
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CROSSREFS
| Cf. A179108, A179387.
Sequence in context: A195002 A203756 A153737 * A064029 A057645 A129060
Adjacent sequences: A078930 A078931 A078932 * A078934 A078935 A078936
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KEYWORD
| nonn
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AUTHOR
| Dean Hickerson (dean.hickerson(AT)yahoo.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 16 2002
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