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%I #19 Sep 08 2022 08:45:58
%S 2,0,0,2,0,0,0,0,0,0,6,0,0,0,0,2,0,0,2,4,0,0,0,0,2,0,0,0,4,0,4,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,2,0,0,0,0,
%U 0,0,2,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0
%N Number of integer solutions to the quartic elliptic curve y^2 = 5*x^4 - 4*n.
%C The quintic x^5+n*x+m is reducible into cubic and quadratic factors if and only a(n) != 0.
%e We have following parametrization: (X^3 - d*X^2 + (d^2 - e)*X + (2*d*e - d^3))*(X^2 + d*X + e) = -d^3*e + 2*d*e^2 + (-d^4 + 3*d^2*e - e^2)*X + X^5.
%e Solving the equation (-d^4 + 3*d^2*e - e^2) = n for e we have e=(3*d^2 +/-sqrt(5*d^4 - 4*n))/2. So 5*d^4 - 4*n must be a perfect square (then y^2=5*x^4-4*n has at least one integer solution).
%o (Magma) [IntegralQuarticPoints([5,0,0,0,-4*n]) : n in [1..55]];
%Y Cf. A193524, A193528.
%K nonn
%O 1,1
%A _Artur Jasinski_, Jul 29 2011