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

1,1

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

Table of n, a(n) for n=1..13.

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.

Prog

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

Crossrefs

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

Sequence in context: A288627 A232096 A250030 * A370060 A344893 A338946

Adjacent sequences: A316453 A316454 A316455 * A316457 A316458 A316459

Keyword

sign,fini,full

Author

Tomohiro Yamada, Jul 04 2018

Status

approved