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A193584
Numbers k such that quartic elliptic curve y^2 = 5x^4 + 4k have integer solutions.
3
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
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
KEYWORD
nonn
AUTHOR
Artur Jasinski, Jul 31 2011
EXTENSIONS
Terms a(32) onward from Max Alekseyev, Apr 26 2015
STATUS
approved