A258928 a(n) = number of integral points on the elliptic curve y^2 = x^3 - (n^2)*x + 1, considering only nonnegative values of y.

3, 6, 11, 9, 15, 13, 14, 17, 26, 12, 12, 11, 12, 19, 20, 11, 19, 36, 12, 17, 16, 11, 19, 16, 15, 27, 17, 17, 18, 16, 12, 15, 17, 11, 12, 11, 28, 16, 12, 11, 15, 24, 27, 11, 17, 12, 26, 15, 17, 15, 12, 15, 17, 27, 12, 14, 16, 15, 16, 24, 12, 41, 17, 16, 12, 11, 17, 16, 16, 15, 23, 15, 16, 20, 15

Offset

0,1

Comments

For n>3, the number of integral points on y = x^3 - (n^2)*x + 1 is at least 11. These 11 points correspond to the solutions x = {-1, 0, n, -n, n + 2, -n + 2, n^2 - 1, n^2 - 2n + 2, n^2 + 2n + 2, n^4 + 2n, n^4 - 2n}.

Links

Robert Israel, Table of n, a(n) for n = 0..209

Example

a(0) = 3 because the integer points on y^2 = x^3 + 1 are (-1, 0), (0, 1), and (2, 3).

Prog

(Sage)
def f(n):
  R.<x, y> = QQ[]
  E = EllipticCurve(y^2 - x^3 + n^2*x - 1)
  return len(E.integral_points(both_signs=false))
[f(x) for x in range(40)]  # Robert Israel, Apr 23 2021

Crossrefs

Cf. A081119, A081120, A259191.

Sequence in context: A117128 A006509 A325551 * A144562 A102889 A183543

Adjacent sequences: A258925 A258926 A258927 * A258929 A258930 A258931

Keyword

nonn

Author

Morris Neene, Jun 14 2015

Extensions

More terms from Robert Israel, Apr 23 2021

Status

approved