

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

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), 731741.
Robin Hartshorne, Algebraic Geometry, GTM 52, SpringerVerlag, 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 * A083716 A231820 A010268
Adjacent sequences: A316453 A316454 A316455 * A316457 A316458 A316459


KEYWORD

sign,fini,full


AUTHOR

Tomohiro Yamada, Jul 04 2018


STATUS

approved



