A370308 Values d for the discriminant d^2 = 4p^3 - 27q*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

1,2

Comments

To obtain integer roots from the depressed cubic x^3 - p*x + q = 0, its discriminant 4p^3 - 27q*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(3) = 70 and occurs when (p, q) = (13, 12). The depressed cubic is given as x*3 - 13x + 12 and has roots (-4, 1, 3}. It is the 3rd occurrence of a solution set of 3 integers.

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