Not that radically difficult

This problem is simpler than how it appears.

Let's focus on each term individually.

Let $x = \sqrt{p(p+1) + \sqrt{p(p+1) + ...}}$ . This can be represented as $x = \sqrt{p(p+1) + x}$ .

Solving for x leads to $x^2 = p(p+1) + x$ , which leads to $p+1$ as the only valid solution.

Let $y = \sqrt{p(p-1) - \sqrt{p(p-1) - ...}}$ . This can be represented as $y = \sqrt{p(p-1) - y}$ .

Solving for y leads to $y^2 = p(p-1) - y$ , which leads to $p-1$ as the only valid solution.

Therefore the answer is $\boxed{2}$ .