Existence, Uniqueness and Counterexamples · Hard problem

Pigeonhole shows that some power $3^n$ with $n \ge 1$ ends in the digits $001$: two of $3^1, \ldots, 3^{1001}$ agree mod $1000$, and dividing gives $3^{j - i} \equiv 1$. What is the SMALLEST such $n$?