A193587 Numbers k such that the quartic elliptic curve y^2 = 5x^4 - 4k has integer solutions.

1, 4, 11, 16, 19, 20, 25, 29, 31, 45, 59, 64, 71, 79, 81, 89, 95, 99, 101, 124, 131, 139, 151, 169, 176, 179, 181, 191, 199, 211, 220, 229, 239, 245, 251, 256, 271, 275, 284, 295, 304, 311, 316, 319, 320, 324, 349, 359, 361, 369, 379, 395, 400, 401, 439, 451

Offset

1,2

Comments

For these numbers k there exists an integer m such that the quintic trinomial x^5+k*x+m factors as a cubic times a quadratic.

Positive numbers of the form -d^4 + 3 d^2 e - e^2.

Links

Table of n, a(n) for n=1..56.

Formula

Complement to A193533.

Mathematica

aa = {}; Do[Do[k = -d^4 + 3 d^2 e - e^2; If[k > 0, AppendTo[aa, k ]], {d, -100, 100}], {e, -100, 100}]; Take[Union[aa], 100]

Crossrefs

Cf. A193524, A193528, A193531, A193533, A193584.

Sequence in context: A091391 A135105 A330338 * A261155 A248104 A072423

Adjacent sequences: A193584 A193585 A193586 * A193588 A193589 A193590

Keyword

nonn

Author

Artur Jasinski, Jul 31 2011

Status

approved