Review question

# When is $4^n-1$ prime? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5927

## Solution

For how many positive integers $n$ is $4^n-1$ a prime number?

We recall that a prime number has exactly two factors, $1$ and itself.

(The number $1$ is not considered to be a prime number these days.)

We can factorise $4^n - 1$ as $4^n-1 = 2^{2n} - 1 = (2^n - 1)(2^n + 1).$

Now $2^n-1 < 2^n+1$, so for $4^n - 1$ to be prime, we must have $2^n-1=1$, which means $n$ must be $1$.

Checking for $n = 1$, we see that $4^1-1=3$ is indeed prime.

Thus, there is exactly one value of $n$ for which $4^n-1$ is a prime number.