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

%I A081121 #38 Feb 16 2025 08:32:48
%S A081121 3,5,6,9,10,12,14,16,17,21,22,24,29,30,31,32,33,34,36,37,38,41,42,43,
%T A081121 46,50,51,52,57,58,59,62,65,66,68,69,70,73,75,77,78,80,82,84,85,86,88,
%U A081121 90,91,92,93,94,96,97,98,99
%N A081121 Numbers k such that Mordell's equation y^2 = x^3 - k has no integral solutions.
%C A081121 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.
%C A081121 This is the complement of A106265. - _M. F. Hasler_, Oct 05 2013
%C A081121 Numbers k such that A081120(k) = 0. - _Charles R Greathouse IV_, Apr 29 2015
%D A081121 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
%H A081121 T. D. Noe, <a href="/A081121/b081121.txt">Table of n, a(n) for n = 1..7757</a> (from Gebel, 3136 and 6789 removed by _Seth A. Troisi_, May 20 2019)
%H A081121 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 A081121 J. Gebel, A. Petho and G. Zimmer, <a href="https://doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Mathematica 110 (3) (1998), 335-367. <a href="http://www.ams.org/mathscinet-getitem?mr=1602064">MR1602064</a>.
%H A081121 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%t A081121 m = 99; f[_List] := (xm = 2 xm; ym = Ceiling[xm^(3/2)];
%t A081121 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 *)
%Y A081121 Cf. A054504, A081120, A106265.
%K A081121 nice,nonn
%O A081121 1,1
%A A081121 _T. D. Noe_, Mar 06 2003