A173202 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)

0, 5, 58, 207, 500, 985, 1710, 2723, 4072, 5805, 7970, 10615, 13788, 17537, 21910, 26955, 32720, 39253, 46602, 54815, 63940, 74025, 85118, 97267, 110520, 124925, 140530, 157383, 175532, 195025, 215910, 238235, 262048, 287397, 314330, 342895

Offset

1,2

Comments

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.

References

T. Apostol, Introduction to Analytic Number Theory, Springer, 1976

D. Duverney, Theorie des nombres (2e edition), Dunod, 2007, p.151

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory 4 (1972), 107-117.

Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k (solutions for 0<k<1008).

School of Mathematics and Statistics, University of St Andrews, Louis Joel Mordell.

Eric Weisstein's World of Mathematics, Mordell Curve.

D. J. Wright, Mordell's Equation.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

y = a*(8*a^2 - 3).

a(n) = sqrt(A000466(n)^3 - A080663(n)). - Artur Jasinski, Nov 26 2011

From Colin Barker, Apr 26 2012: (Start)

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

G.f.: x^2*(5 + 38*x + 5*x^2)/(1 - x)^4. (End)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012

E.g.f.: exp(x)*(5*x + 24*x^2 + 8*x^3). - Stefano Spezia, Dec 04 2018

Example

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

Maple

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A068003 A295902 A290343 * A104099 A334698 A129897

Adjacent sequences: A173199 A173200 A173201 * A173203 A173204 A173205

Keyword

nonn,easy

Author

Michel Lagneau, Feb 12 2010

Status

approved