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, 12813608766102806, 23415546067124892, 38115991067861271

Offset: 1

Author

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

Keywords

• nonn

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]

Examples

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

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

• Frank A. Stevenson, Table of n, a(n) for n = 1..54
• S. Aanderaa, L. Kristiansen, and H. K. Ruud, Search for good examples of Hall's conjecture, Math. Comp. 87 (2018), 2903-2914.
• 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.

Crossrefs

Cf. A179108, A179387, A200216.

Programs

Mathematica

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