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Numbers k such that the quartic elliptic curve y^2 = 5x^4 - 4k has integer solutions.
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%I #15 Nov 16 2017 10:06:03

%S 1,4,11,16,19,20,25,29,31,45,59,64,71,79,81,89,95,99,101,124,131,139,

%T 151,169,176,179,181,191,199,211,220,229,239,245,251,256,271,275,284,

%U 295,304,311,316,319,320,324,349,359,361,369,379,395,400,401,439,451

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

%C 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.

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

%F Complement to A193533.

%t 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]

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

%K nonn

%O 1,2

%A _Artur Jasinski_, Jul 31 2011