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

1,1

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

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 *)

Crossrefs

Cf. A081119, A081121.

Sequence in context: A164018 A156793 A081715 * A190612 A270431 A315832

Adjacent sequences: A054501 A054502 A054503 * A054505 A054506 A054507

Keyword

nonn,nice

Author

N. J. A. Sloane, Apr 08 2000

Extensions

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

Status

approved