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A316456 Complete list of solutions to y^2 = x^3 - 7x + 10; sequence gives x values. 1
-3, -2, -1, 1, 2, 3, 5, 9, 13, 31, 41, 67, 302 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
sign,fini,full
AUTHOR
Tomohiro Yamada, Jul 04 2018
STATUS
approved

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Last modified July 20 08:46 EDT 2024. Contains 374442 sequences. (Running on oeis4.)