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

Author

Morris Neene, Jun 14 2015

Keywords

• nonn

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}.

Examples

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

Links

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

Crossrefs

Cf. A081119, A081120, A259191.

Programs

Sage

def f(n):
  R. = 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

Extensions

More terms from Robert Israel, Apr 23 2021