Problem Description:

Being a some constant, further assume that we
are in a factor ring (basically all operations modulo some sumber p). Note, that the division below is a multiplication by the modular inverse.
You always have to start with x=9.
Consider the following recursive formula:

Code:
new_x = (x²-1)² / (4*x*(x²+a*x+1))

How often do you have to perform this operation to get a specific x (basically getting the new_x and feeding it back into the formula to get another new_x, and so on)?
Note: You can start multiple such chains beginning at x=9, and add the resulting x values
using the addition algorithm from http://en.wikipedia.org/wiki/Montgomery_curve (Montgomery arithmetic section).
Note, that the x value, is the value you get at the end of such calculation-chain, and the z value is always 1.