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
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, Feb 12 2010
STATUS
approved