I think there may be no solution based on the following.
f(x) = x^3 + ax^2 + bx + c
f(a) = a^3 + a^3 + ab + c = a^3
a^3 + ab + c = 0
f(b) = b^3 + ab^2 + b^2 + c = b^3
ab^2 + b^2 + c = 0
a^3 + ab = ab^2 + b^2
a^3 + ab - ab^2 - b^2 = 0 which works when a=b, but the coefficients have to be distinct non-zero integers.
Just checking with a spreadsheet, it looks like there are no other solutions to this last equation other than a=b.
Posted by Larry
on 2020-11-02 13:31:59