Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n - 1)^n + (p_n + 1)^n$ is divided by $p_n^2$.

For example, when $n = 3$, $p_3 = 5$, and $4^3 + 6^3 = 280 \equiv 5 \mod 25$.

The least value of $n$ for which the remainder first exceeds $10^9$ is $7037$.

Find the least value of $n$ for which the remainder first exceeds $10^{10}$.