Primes of form Power Less One

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Theorem

Let $m, n \in \N_{>0}$ be natural numbers.

Let $m^n - 1$ be prime.


Then $m = 2$ and $n$ is prime.


Proof

First we note that by Integer Less One divides Power Less One:

$\paren {m - 1} \divides \paren {m^n - 1}$

where $\divides$ denotes divisibility.

Thus $m^n - 1$ is composite for all $m \in \Z: m > 2$.


Let $m = 2$, and consider $2^n - 1$.


Suppose $n$ is composite.

Then $n = r s$ where $r, s \in \Z_{> 1}$.

Then by the corollary to Integer Less One divides Power Less One:

$\paren {2^r - 1} \divides \paren {2^{r s} - 1}$


Thus if $n$ is composite, then so is $2^n - 1$.

So $2^n - 1$ can be prime only when $n$ is prime.

$\blacksquare$


Also see

Primes of the form $2^n - 1$ are called Mersenne primes.

They are particularly interesting because there is a convenient algorithm (the Lucas-Lehmer Test) which can determine the primality of such a number with high computational efficiency.

Hence, at the present time, the largest primes known are Mersenne.


Historical Note

The proof that if $n$ is composite, then so is $2^n - 1$, is historically attributed to Cataldi, who gave it in $1603$.

Marin Mersenne was aware of this result, but took it further to ask the question as to which numbers of the form $2^p - 1$ are prime when $p$ is prime.


Sources