## How to Factorize 2^n-1

If is composite, we may write where .

Then,

The key point is that both factors are greater than 1.

## Example

If , we can see that .

## How to Remember this Result

The hard part is remembering this identity: . It is essentially a “telescoping sum”, and you can prove it by expanding the right hand side and cancelling all like terms.

We can also view it as a Geometric Series as follows: is the sum of a Geometric Progression with first term 1, and common ratio , for a total of terms. That is,

. Multiplying both sides by gives the boxed result.

## Applications

Primes of the form are known as Mersenne Primes. Using the result above, we can show that if is a prime, then is a prime. (We prove the contrapositive: If is composite, then we show that is also composite.)