Here are two very similar questions about polynomials that I feel may have deeper roots (excuse pun). a) Does anyone know of any interesting theory related to them that I could read up upon? b) How would one start solving them? Here are the problems: 1) Show that there are infinitely many polynomials p with integer coefficients such that P(x^2 - 1) = (P(x))^2 - 1, P(0) =0 . 2) Are there real polynomials p satisfying P(x^2 - 1) = (P(x))^2 + 1 for all x? If so, determine what they look like. Observe the plus sign at the end of the second problem.