Consecutive square-free numbers and square-free primitive roots

Let H be a positive integer and let p be an odd prime. For 1 <= x, y <= H, we study the distribution of consecutive square-free numbers of the forms x(2) + y(2) + 1, x(2) + y(2) + 2 and xy + 1, xy + 2. In addition, we study the distribution of consecutive square-free primitive roots modulo p of the forms x(2) + y(2) + 1, x(2) + y(2) + 2, xy + 1, xy + 2, and x(2) + 1, x(2) + 2, respectively.