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.

0, 20, 0, 70, 56, 162, 0, 160, 308, 110, 324, 520, 0, 286, 560, 810, 182, 540, 880, 1190, 0, 448, 884, 1296, 1672, 272, 810, 1330, 1820, 0, 646, 2268, 1280, 1890, 2464, 380, 1134, 2990, 1870, 2576, 3240, 0, 880, 1748, 2592, 3850, 3400, 506, 1512

Offset: 1

Author

Frank M Jackson, Feb 14 2024

Keywords

• nonn

Comments

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.

Examples

			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}.
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}.
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}.
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}.
		

Links

• Wikipedia, Discriminant - Degree 3.

Crossrefs

Cf. A082375.

Programs

Mathematica

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