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

%I A054504 #32 Feb 16 2025 08:32:42
%S A054504 6,7,11,13,14,20,21,23,29,32,34,39,42,45,46,47,51,53,58,59,60,61,62,
%T A054504 66,67,69,70,74,75,77,78,83,84,85,86,87,88,90,93,95,96,102,103,104,
%U A054504 109,110,111,114,115,116,118,123,124,130,133,135,137,139,140,146,147,149,153,155
%N A054504 Numbers n such that Mordell's equation y^2 = x^3 + n has no integral solutions.
%C A054504 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
%C A054504 Numbers n such that A081119(n) = 0. - _Charles R Greathouse IV_, Apr 29 2015
%D A054504 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 192.
%D A054504 J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.
%H A054504 T. D. Noe, <a href="/A054504/b054504.txt">Table of n, a(n) for n = 1..6603</a> (from Gebel)
%H A054504 Pantelis Andreou, Stavros Konstantinidis, and Taylor J. Smith, <a href="https://arxiv.org/abs/2403.08707">Improved Randomized Approximation of Hard Universality and Emptiness Problems</a>, arXiv:2403.08707 [cs.DS], 2024. See p. 16.
%H A054504 Ryan D'Mello, <a href="http://arxiv.org/abs/1410.0078">Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves</a>, arXiv preprint arXiv:1410.0078, 2014
%H A054504 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H A054504 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%t A054504 m = 155; f[_List] := ( xm = 2 xm; ym = Ceiling[xm^(3/2)];
%t A054504 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 *)
%Y A054504 Cf. A081119, A081121.
%K A054504 nonn,nice
%O A054504 1,1
%A A054504 _N. J. A. Sloane_, Apr 08 2000
%E A054504 Apostol gives all values of n < 100. Extended by _David W. Wilson_, Sep 25 2000