A054504 Numbers n such that Mordell's equation y^2 = x^3 + n has no integral solutions.

6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, 45, 46, 47, 51, 53, 58, 59, 60, 61, 62, 66, 67, 69, 70, 74, 75, 77, 78, 83, 84, 85, 86, 87, 88, 90, 93, 95, 96, 102, 103, 104, 109, 110, 111, 114, 115, 116, 118, 123, 124, 130, 133, 135, 137, 139, 140, 146, 147, 149, 153, 155

Offset: 1

Author

N. J. A. Sloane, Apr 08 2000

Keywords

• nonn
• nice

Comments

Mordell's equation has a finite number of integral solutions for all nonzero n. Gebel computes the solutions for n < 10^5. Sequence A081121 gives n for which there are no integral solutions to y^2 = x^3 - n. See A081119 for the number of integral solutions to y^2 = x^3 + n. - T. D. Noe, Mar 06 2003

Numbers n such that A081119(n) = 0. - Charles R Greathouse IV, Apr 29 2015

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 192.
• J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

Links

• T. D. Noe, Table of n, a(n) for n = 1..6603 (from Gebel)
• Pantelis Andreou, Stavros Konstantinidis, and Taylor J. Smith, Improved Randomized Approximation of Hard Universality and Emptiness Problems, arXiv:2403.08707 [cs.DS], 2024. See p. 16.
• Ryan D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078, 2014
• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
• Eric Weisstein's World of Mathematics, Mordell Curve

Crossrefs

Cf. A081119, A081121.

Programs

Mathematica

m = 155; f[_List] := ( xm = 2 xm; ym = Ceiling[xm^(3/2)];
Complement[Range[m], Outer[Plus, Range[0, ym]^2, -Range[-xm, xm]^3] //Flatten //Union]); xm=10; FixedPoint[f, {}] (* Jean-François Alcover, Apr 28 2011 *)

Extensions

Apostol gives all values of n < 100. Extended by David W. Wilson, Sep 25 2000