MathNet 2014 Question 82

Let $p^n$ be a prime power. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i \in \{0, 1, \dots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that
$$
p^n \mid (a_1a_2 + a_3a_4 + 1).
$$