A173202 - OEIS

%I #41 Mar 06 2023 08:46:04

%S 0,5,58,207,500,985,1710,2723,4072,5805,7970,10615,13788,17537,21910,

%T 26955,32720,39253,46602,54815,63940,74025,85118,97267,110520,124925,

%U 140530,157383,175532,195025,215910,238235,262048,287397,314330,342895

%N Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466)

%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 an 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, Theorie des nombres (2e edition), Dunod, 2007, p.151

%H Vincenzo Librandi, <a href="/A173202/b173202.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 Helmut Richter, <a href="http://www.lrz-muenchen.de/~hr/numb/mordell.html">Solutions of Mordell's equation y^2 = x^3 + k</a> (solutions for 0<k<1008).

%H School of Mathematics and Statistics, University of St Andrews, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Mordell.html">Louis Joel Mordell</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>.

%H D. J. Wright, <a href="http://cauchy.math.okstate.edu/~wrightd/4713/nt_essay/node2.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 a(n) = sqrt(A000466(n)^3 - A080663(n)). - _Artur Jasinski_, Nov 26 2011

%F From _Colin Barker_, Apr 26 2012: (Start)

%F a(n) = 8*n^3 - 24*n^2 + 21*n - 5.

%F G.f.: x^2*(5 + 38*x + 5*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

%F E.g.f.: exp(x)*(5*x + 24*x^2 + 8*x^3). - _Stefano Spezia_, Dec 04 2018

%e With a=3, x = 35 and y = 207, and then 207^2 = 35^2 - 26.

%p for a from 0 to 100 do : z := evalf(a*(8*a^2 - 3)) : print (z) :od :

%t CoefficientList[Series[x*(5+38*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 02 2012 *)

%t CoefficientList[Series[E^x (5 x + 24 x^2 + 8 x^3), {x, 0, 40}], x]*Table[n!, {n, 0, 40}] (* _Stefano Spezia_, Dec 04 2018 *)

%o (Magma) I:=[0, 5, 58, 207]; [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

%Y Diophantine equations: see also Pellian equation: (A081233, A081234), (A081231, A082394), (A081232, A082393); Mordell equation: A053755, A173200; Diophantine equations: A006452, A006451, A006454.

%K nonn,easy

%O 1,2

%A _Michel Lagneau_, Feb 12 2010