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A054504
Numbers n such that Mordell's equation y^2 = x^3 + n has no integral solutions.
45
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.
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
Sequence in context: A164018 A156793 A081715 * A190612 A270431 A315832
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