Mersenne number


Are there infinite number of Mersenne Primes? ★★★★

Author(s):

\begin{conjecture} A Mersenne prime is a Mersenne number \[ M_n = 2^p - 1 \] that is prime.

Are there infinite number of Mersenne Primes? \end{conjecture}

Keywords: Mersenne number; Mersenne prime

Are all Mersenne Numbers with prime exponent square-free? ★★★

Author(s):

\begin{conjecture} Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free? \end{conjecture}

Keywords: Mersenne number

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