A078933 Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.

2, 5234, 8158, 93844, 367806, 421351, 720114, 939787, 28187351, 110781386, 154319269, 384242766, 390620082, 3790689201, 65589428378, 952764389446, 12438517260105, 35495694227489, 53197086958290, 5853886516781223

Offset

1,1

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. [Artur Jasinski, Nov 04 2011]

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.

Links

Table of n, a(n) for n=1..20.

Ismael Jimenez Calvo, Marshall 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, and 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.

R. D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078 [math.NT], 2014.

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.

Example

|5234^3 - 378661^2| = 17 < sqrt(5234), so 5234 is in the sequence.

Mathematica

For[x=1, True, x++, If[Abs[x^3-Round[Sqrt[x^3]]^2] < Sqrt[x] && !IntegerQ[Sqrt[x]], Print[x]]]

Crossrefs

Cf. A179108, A179387, A200216.

Sequence in context: A232712 A262059 A153737 * A064029 A057645 A129060

Adjacent sequences: A078930 A078931 A078932 * A078934 A078935 A078936

Keyword

nonn,more

Author

Dean Hickerson and Robert G. Wilson v, Dec 16 2002

Status

approved