A316456 Complete list of solutions to y^2 = x^3 - 7x + 10; sequence gives x values.

-3, -2, -1, 1, 2, 3, 5, 9, 13, 31, 41, 67, 302

Offset: 1

Author

Tomohiro Yamada, Jul 04 2018

Keywords

• sign
• fini
• full

Comments

Bremner and Tzanakis showed that the list of solutions is complete.

The elliptic curve given by this equation has rank 2 over the rationals with generators (1, 2) and (2, 2).

Since there exist two integer points (x, y) and (x, -y) for each x in the sequence (we can easily see that y <> 0 for such an x), this elliptic curve has exactly 26 integer points.

Links

• Andrew Bremner and Nicholas Tzanakis, Integer points on y^2 = x^3 - 7x + 10, Math. Comp. 41 (1983), 731-741.
• Robin Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, Chapter IV, Exercise 4.18.

Crossrefs

Cf. A029728 (y^2 = x^3 + 17), A047694 (y^2 = C(x,0) + C(x,1) + C(x,2) + C(x,3))

Programs

SageMath

EllipticCurve([0,0,0,-7,10]).integral_points()