A028943 Denominator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

1, 1, 1, 1, 8, 1, 27, 125, 343, 64, 12167, 24389, 205379, 2146689, 30959144, 274625, 3574558889, 50202571769, 553185473329, 4302115807744, 578280195945297, 1469451780501769, 238670664494938073, 13528653463047586625

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives numerators of the x_n. - N. J. A. Sloane, Jan 27 2022

Examples

			5P = (1/4, -5/8).
		

References

• A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..173
• B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Crossrefs

Cf. A028940, A028941, A028942.

Programs

PARI

See A028940.

Formula

P=(0, 0), 2P=(1, 0), if kP=(a, b) then (k+1)P=(a'=(b^2-a^3)/a^2, b'=-1-b*a'/a).