A173200 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0,1,2, ... (solutions x are given by A053755).

0, 11, 70, 225, 524, 1015, 1746, 2765, 4120, 5859, 8030, 10681, 13860, 17615, 21994, 27045, 32816, 39355, 46710, 54929, 64060, 74151, 85250, 97405, 110664, 125075, 140686, 157545, 175700, 195199, 216090, 238421, 262240, 287595, 314534, 343105

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 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, Théorie 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.

John J. O'Connor and Edmund F. Robertson, Louis Joel Mordell

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

Eric Weisstein's World of Mathematics, Mordell Curve

David 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).

From Colin Barker, Apr 26 2012: (Start)

a(n) = 8*n^3 - 24*n^2 + 27*n - 11.

G.f.: x^2*(11 + 26*x + 11*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

Example

With a=3, x =37 and y = 225, and then 225^2 = 37^2 - 28.

Maple

for a from 0 to 150 do : z := evalf(a*(8*a^2 + 3)) : print (z) :od :

Mathematica

CoefficientList[Series[x*(11+26*x+11*x^2)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 02 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 11, 70, 225}, 40] (* Harvey P. Dale, Dec 21 2016 *)

Prog

(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
(Python) for n in range(1, 20): print(8*n**3 - 24*n**2 + 27*n - 11, end=', ') # Stefano Spezia, Dec 05 2018

Crossrefs

Sequence in context: A236320 A211050 A295074 * A071746 A205812 A162568

Adjacent sequences: A173197 A173198 A173199 * A173201 A173202 A173203

Keyword

nonn,easy

Author

Michel Lagneau, Feb 12 2010

Status

approved