A193584 Numbers k such that quartic elliptic curve y^2 = 5x^4 + 4k have integer solutions.

1, 4, 5, 9, 11, 16, 19, 25, 29, 31, 36, 41, 44, 49, 55, 59, 61, 64, 71, 79, 80, 81, 89, 100, 101, 109, 121, 124, 131, 139, 144, 149, 155, 164, 169, 171, 176, 181, 191, 196, 205, 209, 211, 225, 229, 236, 239, 241, 251, 256, 269, 271, 275, 279, 289, 304, 305, 316, 319, 324, 331, 341, 355, 356, 361, 379, 380, 400, 405, 409, 419, 421, 439, 441, 449, 451, 461, 464, 475

Offset

1,2

Comments

For these k, there exist an integer m such that quintic trinomial x^5-k*x+m is reducible into cubic and quadratic factors.

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

The curve is equivalent to Y^2 = 5*X^3 + 4k*X, where Y=xy and X=x^2. - Max Alekseyev, Apr 26 2015

Links

Table of n, a(n) for n=1..79.

Formula

Complement to A193528

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]

Prog

(Magma) for k:=1 to 1000 do if IntegralQuarticPoints([5, 0, 0, 0, 4*k]) ne [] then print(k); end if; end for; /* Max Alekseyev, Apr 26 2015 */

Crossrefs

Cf. A193524, A193528, A193531, A193533.

Sequence in context: A166562 A031363 A118142 * A155149 A024821 A059610

Adjacent sequences: A193581 A193582 A193583 * A193585 A193586 A193587

Keyword

nonn

Author

Artur Jasinski, Jul 31 2011

Extensions

Terms a(32) onward from Max Alekseyev, Apr 26 2015

Status

approved