A303615 Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.

-29, -25, -20, -14, -5, 5, 14, 16, 20, 25, 49, 70, 79, 130, 250, 305, 400, 695, 1555, 1645, 18895

Offset

1,1

Comments

This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points in this curve.

This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).

Links

Table of n, a(n) for n=1..21.

Roelof J. Stroeker and Benjamin M. M. de Weger, Elliptic binomial diophantine equations, Math. Comp. 68 (1999), 1257-1281.

Example

a(6) = 5: 5^3 - 525 * 5 + 10156 = 7656 = 88 * 87.

Crossrefs

Cf. A029728 (the complete list of solutions x to y^2=x^3+17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).

Sequence in context: A088400 A040814 A307129 * A291492 A256441 A261310

Adjacent sequences: A303612 A303613 A303614 * A303616 A303617 A303618

Keyword

sign,fini,full

Author

Tomohiro Yamada, May 29 2018

Status

approved