

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
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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^327q^2)^(1/2)], (sol=Solve[x^3p*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



KEYWORD

nonn


AUTHOR



STATUS

approved



