My first problem

Number Theory Level 3

$\large{T_m=\underbrace{17^{17^{\cdot^{\cdot^{\cdot^{17}}}}}}_{m \text{ times}}}$

Find the smallest possible value of $n$ such that the numbers $T_n,T_{n+1},T_{n+2},\ldots$ all have the same last (rightmost) 2017 digits.

Bonus: Generalize to the last $k$ digits.