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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
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Frank M Jackson, Feb 14 2024
STATUS
approved

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Last modified July 19 17:03 EDT 2024. Contains 374410 sequences. (Running on oeis4.)