= Mersenne number
{c}
{title2=$M_k=2^k-1,\quad k\geq1$}
A <Mersenne number> has the form $M_k=2^k-1$ for a <positive integer> $k$. Its <binary representation> consists of $k$ ones. For example, $M_4=15$ is composite, while $M_5=31$ is <prime>. If $k=ab$ with $a,b>1$, the <finite geometric series> factorization
$$
2^{ab}-1=(2^a-1)(1+2^a+\cdots+2^{a(b-1)})
$$
has two factors greater than one. Consequently a <Mersenne prime> requires a <prime> exponent, but $M_{11}=23\cdot89$ shows that this condition alone is insufficient.
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