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%I #34 Sep 08 2022 08:45:50
%S 0,11,70,225,524,1015,1746,2765,4120,5859,8030,10681,13860,17615,
%T 21994,27045,32816,39355,46710,54929,64060,74151,85250,97405,110664,
%U 125075,140686,157545,175700,195199,216090,238421,262240,287595,314534,343105
%N Solutions y of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0,1,2, ... (solutions x are given by A053755).
%C For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466.
%D T. Apostol, Introduction to Analytic Number Theory, Springer, 1976
%D D. Duverney, Théorie des nombres (2e edition), Dunod, 2007, p.151
%H Vincenzo Librandi, <a href="/A173200/b173200.txt">Table of n, a(n) for n = 1..1000</a>
%H W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, <a href="https://doi.org/10.1016/0022-314X(72)90058-3">The diophantine equation y^2 + k = x^3</a>, J. Number Theory 4 (1972), 107-117.
%H John J. O'Connor and Edmund F. Robertson, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Mordell.html">Louis Joel Mordell</a>
%H Helmut Richter, <a href="http://hr.userweb.mwn.de/numb/mordell.html">Solutions of Mordell's equation y^2 = x^3 + k</a> (solutions for 0<k<1008)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%H David J. Wright, <a href="http://cauchy.math.okstate.edu/~wrightd/4713/nt_essay/node5.html">Mordell's Equation</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F y = a*(8*a^2 + 3).
%F From _Colin Barker_, Apr 26 2012: (Start)
%F a(n) = 8*n^3 - 24*n^2 + 27*n - 11.
%F G.f.: x^2*(11 + 26*x + 11*x^2)/(1 - x)^4. (End)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jul 02 2012
%e With a=3, x =37 and y = 225, and then 225^2 = 37^2 - 28.
%p for a from 0 to 150 do : z := evalf(a*(8*a^2 + 3)) : print (z) :od :
%t CoefficientList[Series[x*(11+26*x+11*x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 02 2012 *)
%t LinearRecurrence[{4,-6,4,-1},{0,11,70,225},40] (* _Harvey P. Dale_, Dec 21 2016 *)
%o (Magma) I:=[0, 11, 70, 225]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jul 02 2012
%o (Python) for n in range(1,20): print(8*n**3 - 24*n**2 + 27*n - 11, end=', ') # _Stefano Spezia_, Dec 05 2018
%K nonn,easy
%O 1,2
%A _Michel Lagneau_, Feb 12 2010
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