TITLE: When does $2^{n}$ divide $3^{n}-1$
QUESTION [7 upvotes]: The title says it all. For what natural numbers $n$ does $2^n$ divide $3^n - 1$? By substituting values I can see that this happens for $n = 1, 2, 4$ but are there more? I am not able to prove/disprove.

REPLY [5 votes]: If $n$ is odd then $3^n-1 \equiv 2\pmod 4$. Otherwise the number of factors of $2$ dividing $3^n-1$ is exactly $2+\upsilon_2(n)$by litfing the exponent, which is at most $2+\ln_2(n)$. Hence we need
$$n \le 2+\ln_2(n) \implies n<5.$$
Therefore the unique solutions are $1,2,4$.