A081121 Numbers k such that Mordell's equation y^2 = x^3 - k has no integral solutions.

3, 5, 6, 9, 10, 12, 14, 16, 17, 21, 22, 24, 29, 30, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 46, 50, 51, 52, 57, 58, 59, 62, 65, 66, 68, 69, 70, 73, 75, 77, 78, 80, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 97, 98, 99

Offset: 1

Author

T. D. Noe, Mar 06 2003

Keywords

• nice
• nonn

Comments

Mordell's equation has a finite number of integral solutions for all nonzero k. Gebel computes the solutions for k < 10^5. Sequence A054504 gives k for which there are no integral solutions to y^2 = x^3 + k. See A081120 for the number of integral solutions to y^2 = x^3 - n.

This is the complement of A106265. - M. F. Hasler, Oct 05 2013

Numbers k such that A081120(k) = 0. - Charles R Greathouse IV, Apr 29 2015

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

Links

• T. D. Noe, Table of n, a(n) for n = 1..7757 (from Gebel, 3136 and 6789 removed by _Seth A. Troisi_, May 20 2019)
• 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]
• J. Gebel, A. Petho and G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367. MR1602064.
• Eric Weisstein's World of Mathematics, Mordell Curve

Crossrefs

Cf. A054504, A081120, A106265.

Programs

Mathematica

m = 99; 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 29 2011 *)