# When is $(2^a-1)$ a power of 3

I am looking to characterize the values of $a\in\mathbb{Z}$ for which $(2^a-1)$ is an integral power of 3. In particular, are there any besides $a=1,2$? Any positive/negative results would be much appreciated. Thanks!

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Suppose $3^b=2^a-1$ for some integers $b,a$ with $a>2$. Reducing mod $8$, we have $3^b\equiv-1$. But the only powers of $3$ mod $8$ are easily seen to be $1$ and $3$, so we have a contradiction.