Answer
Let $n=7$. Then $2^{n}-1=2^{7}-1=128-1=127$. Since $127$ is prime, we conclude that there are integers $n$ such that $n\gt5$ and $2^{n}-1$ is prime.
Work Step by Step
This problem is deceptively simple. Although it is fairly straightforward to show that no composite number $n$ will work, showing that a prime $n$ produces another prime $2^{n}-1$ is considered extremely difficult, and in general requires proving that the number has no positive factors other than one and itself. In this example, we would either have to show by hand that $127$ is not divisible by $2$, or by $3$, or by $5$, or by $7$, etc., or we would have to appeal to more sophisticated algorithms that are beyond the scope of this chapter. For those who are interested, primes of the form $2^{n}-1$ are called "Mersenne primes." Finding large Mersenne primes is a difficult task and a lucrative area of modern number theory and cryptography.
