A370308 - OEIS

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..49.

Wikipedia, Discriminant - Degree 3.

EXAMPLE

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}.

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

CROSSREFS

Cf. A082375.

Sequence in context: A365912 A324274 A070708 * A200430 A084029 A343329

Adjacent sequences: A370305 A370306 A370307 * A370309 A370310 A370311

KEYWORD

nonn

AUTHOR

Frank M Jackson, Feb 14 2024

STATUS

approved