Then z_{n+2} − Z_{2} can be replaced by its equal value of z_{n+1}^{w} − Z_{1}^{w}. In the second term of the product z_{n+1} − Z_{1} can be replaced by its
equal value of z_{n}^{w} − Z_{2}^{w}.

Thus in order for |z_{n+2} − Z_{2}| to be less than |z_{n} − Z_{2}|
it is necessary that

Thus the condition to be satisfied for a limit 2-cycle is

|w²(Z_{1}Z_{2})^{w-1}| = 1

The complex numbers whose absolute value is equal to 1 are those given by exp(iφ) for 0≤φ≤2π.
Thus

Z_{1}Z_{2} = (1/w²)exp(iφ/(w-1))

The Case of a Positive Integer Exponent

As stated previously, the values of Z_{1} and Z_{2} are solutions to the equation

Z + Z^{w} − λ = 0

When w is a positive integer the above equation is a monic polynomial. The product of the roots of a monic polynomial
is equal to the constant term in the equation. For w=2 then

Z_{1}Z_{2} = −λ

Also for a monic polynomial the sum of the roots is equal to the coefficient of the term with the next to largest exponent.
For w=2 this is 1 and for w>2 it is equal to 0. For w=2 then