A259191 Number of integral solutions to y^2 = x^3 + n*x^2 + n (with y nonnegative).

3, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 8, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 6, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0

Offset: 1

Author

Morris Neene, Jun 20 2015

Keywords

• nonn

Comments

If n is square there are at least two solutions, corresponding to x = 0 and x = -n. If n = 2^(2k) there are at least three solutions, corresponding to x = 0, x = -2^(2k), and x = 2^(6k-2) + 2^(2k). If n = 2k^2 + 2k, there is at least one solution, corresponding to x = 1.

Crossrefs

Cf. A081119, A081120.

Programs

Sage

for i in range(1,31):
    E=EllipticCurve([0,i,0,0,i])
    print(len(E.integral_points()))