The number 107 is a prime number. Let p=107. For a number a such that p \nmid a let a^{-1} be the unique number 0 \leq a^{-1} \leq p^{2}-1 such that p^{2} \mid a a^{-1}-1. Find the number of positive integers b, 1 \leq b \leq \frac{p^{2}-1}{2} such that there exists a number a, 0 \leq a \leq p^{2}-1 such that p^{2} \mid b^{2}-\left(a+a^{-1}\right).