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A370308 Values d for the discriminant d^2 = 4*p^3 - 27*q^2 of the depressed cubic equation x^3 - p*x + q = 0 that give integer roots using integer coefficients p > 0 and q > 0 for increasing p sorted by p then q.

%I A370308 #15 Mar 29 2025 08:04:34
%S A370308 0,20,0,70,56,162,0,160,308,110,324,520,0,286,560,810,182,540,880,
%T A370308 1190,0,448,884,1296,1672,272,810,1330,1820,0,646,2268,1280,1890,2464,
%U A370308 380,1134,2990,1870,2576,3240,0,880,1748,2592,3850,3400,506,1512
%N A370308 Values d for the discriminant d^2 = 4*p^3 - 27*q^2 of the depressed cubic equation x^3 - p*x + q = 0 that give integer roots using integer coefficients p > 0 and q > 0 for increasing p sorted by p then q.
%C A370308 To obtain integer roots from the depressed cubic x^3 - p*x + q = 0, its discriminant 4*p^3 - 27*q^2 has to be a perfect square but this is not a sufficient condition. At least one root has to be integral as well.
%H A370308 Wikipedia, <a href="https://en.wikipedia.org/wiki/Discriminant#Degree_3">Discriminant - Degree 3</a>.
%e A370308 a(1) = 0 and occurs when (p, q) = (3, 2). The depressed cubic is x^3 - 3*x + 2 and has roots {-2, 1, 1}.
%e A370308 a(2) = 20 and occurs when (p, q) = (7, 6). The depressed cubic is x^3 - 7*x + 6 and has roots {-3, 1, 2}.
%e A370308 a(3) = 0 and occurs when (p, q) = (12, 16). The depressed cubic is x^3 - 12*x + 16 and has roots {-4, 2, 2}.
%e A370308 a(4) = 70 and occurs when (p, q) = (13, 12). The depressed cubic is x^3 - 13*x + 12 and has roots {-4, 1, 3}.
%t A370308 lst = {}; Do[If[IntegerQ[k=(4p^3-27q^2)^(1/2)], (sol=Solve[x^3-p*x+q==0, {x}]; {x1, x2, x3}=x /. sol; If[IntegerQ[x1], AppendTo[lst, k]])], {p, 1, 300}, {q, 1, Sqrt[4 p^3/27]}]; lst
%Y A370308 Cf. A082375.
%K A370308 nonn
%O A370308 1,2
%A A370308 _Frank M Jackson_, Feb 14 2024