Can anyone prove the possibility of a Mersenne Number $2^p-1$ that is a perfect power (i.e. $2^p-1 = a^k$ for integers $(p, a, k) > 1$? There are none known with the conditions mentioned. Thanks for help!


Note that $2^p - a^k = 1$ implies that $2^p$ and $a^k$ are consecutive powers. By Mihailescu's Theorem, the only two consecutive powers of natural numbers are

$$3^2 - 2^3 = 1$$

so such $(p, a, k) > 1$ would not exist.


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