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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.

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Those are the only solutions, as follows from Catalan's Conjecture, proved by Mihailescu.

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  • 3
    $\begingroup$ Talk about overkill... Look at anon's answer instead. $\endgroup$ – Omar Antolín-Camarena Feb 20 '13 at 15:51
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    $\begingroup$ Thanks for letting me know about this wonderful theorem! $\endgroup$ – sai Feb 20 '13 at 16:03
  • $\begingroup$ This is a great example of using a cannon to kill a mosquito, however! $\endgroup$ – Emily Feb 20 '13 at 16:31
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    $\begingroup$ It seems superfluous to invent an additional, small instrument to kill a mosquito, when you already have a cannon that will kill it and many other things. :-) $\endgroup$ – LarsH Feb 20 '13 at 20:18

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