1,1

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.

Ismail 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.

L. V. Danilov, The Diophantine equation x^3 - y^2 = k and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.

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.

J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

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.

Ismael Jimenez Calvo, Hall's conjecture.

Noam D. Elkies, List of integers x,y with x<10^18, 0 < |x^3-y^2| < x^(1/2).

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

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

Sequence in context: A105667 A067668 A153737 this_sequence A064029 A057645 A129060

Adjacent sequences: A078930 A078931 A078932 this_sequence A078934 A078935 A078936

nonn

Dean Hickerson (dean.hickerson(AT)yahoo.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 16 2002