Finite difference

Define sequences $x_i,y_i,i=1,2,\dots$ as follows: $$x_{n+1} = x_n^2 - 20$$ $$y_{n+1} = y_n^2 - 30$$ And define $d_i = x_i - y_i$. Find all possible starting pairs $(x_1,y_1)$ such that the set $\{d_1, d_2, \dots \}$ contains only a finite number of elements. Solution Just the outlines. If $x_1 = \pm 4 , \pm 5$ then $x_n$ is bounded, likewise if $y_1 = \pm 5, \pm 6$ it is bounded. It's easy to show that for other possibilities $x_n,y_n$ are both unbounded, and will be positive except for the first few terms.

If $x_n,y_n$ are bounded then they assume a finite number of values, therefore $d_i$ also assume a finite number of values, at most the combinatorial pairs of their images, but most likely much less. However, if one of them is unbounded, then say $x_n$ is unbounded, EVEN IF $y_n$ is bounded, then $x_n+y_n$ is unbounded. At this point it's easy to show that $x_n+y_n$ is monotonically increasing and unbounded (excxept for the first few terms) because once $x_n$ gets large enough, $y_n$'s fluctuation is not enough to make the sum decrease.

Now, if $x_n-y_n$ is unbounded, then $x_n^2 - y_n^2$ is unbounded. If $x_n - y_n$ is bounded then $x_n^2 - y_n^2$ is unbounded, EXCEPT if $x_n = y_n$ identically. But because they have different recursion formula, they can't always be identical. Meaning they're identical at most every other term. On the terms that they are not identical, $x_n-y_n$ is non-zero, and because $x_n+y_n$ is monotonically increasing and unbounded, then $x_n^2 - y_n^2$ is also unbounded. Therefore $d_n +10$ is also unbounded, so it assumes an infinite nunmber of values