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
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
KEYWORD
nonn
AUTHOR
Frank M Jackson, Feb 14 2024
STATUS
approved