If 2^{n}-1 is a prime number, then n is also a prime number. Note that if 2^{n}-1 is prime then it is called a Mersenne prime.

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## If 2^{n}-1 is prime, then prove that n is prime

**Question:** Prove that if 2^{n}-1 is prime, then n is prime.

**Solution:**

The given statement will be proved by the contrapositive method. So assume that n is not a prime number, that is, n is a composite number. So

n=pq

where p and q are integers greater than 1. Now,

2^{n}-1 = 2^{pq} -1

= (2^{p})^{q} -1

= (2^{p}-1) (2^{p(q-1)} + 2^{p(q-2)} + … +2^{p}+1).

As both 2^{p}-1 and 2^{p(q-1)} + 2^{p(q-2)} + … +2^{p}+1 are greater than 1, we conclude that 2^{n}-1 is a composite number, not a prime number. This contradicts the given fact 2^{n}-1 is prime. So our assumption is wrong. Hence we conclude:

If 2^{n}-1 is a prime, then n is also a prime.

**Read Also:** Is 2 a Prime or a Composite Number?

## Examples

We know that 31 is a prime number.

Note that 31=2^{5}-1.

Here n=5 is a prime number. So 31=2^{5}-1 is an example of a Mersenne prime number.

## FAQs

**Q1: If 2**

^{n}-1 is a prime, then is n prime or composite?Answer: If 2^{n}-1 is a prime, then n is always a prime number.