if $2^{n} - 1$ a prime, show that $n$ is a prime.

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Solution

Suppose that $n$ is composite,
hence can be written $n = a b$ with $a$ and $b > 1$ . Then $2^{n} - 1 = 2^{a b} - 1$
$= (2^{a})^{b} - 1$
$= (2^{a} - 1) ((2^{a})^{b - 1} + (2^{a})^{b - 2} + . . . . . . . + 2^{a} + 1)$
So $2^{n} - 1$ is composite. By contropositive, If $2^{n} - 1$ is prime then $n$ is prime.

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