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

Author

Michel Lagneau, Feb 12 2010

Keywords

• nonn
• easy

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.

Examples

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

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, and D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory, Vol. 4 (1972), pp. 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
• 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).

Crossrefs

Cf. A000466.

Programs

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

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 *)

Python

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

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

E.g.f.: 11 + exp(x)*(-11 + 11*x + 8*x^3). - Elmo R. Oliveira, Aug 15 2025