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 $4^3+6^3=280\equiv 5\mathrm{mod}25$.

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 ${10}^{10}$.