To determine whether \( 2^{55} - 1 \) is a Mersenne prime, we should first identify if \( 55 \) itself is prime. If \( n \) is not prime, \( 2^n - 1 \) cannot be a Mersenne prime.
The number \( 55 \) is not prime since it can be factored into \( 5 \times 11 \). Therefore, \( 2^{55} - 1 \) cannot be a Mersenne prime.
Next, let's use the properties of Mersenne numbers to find a factor. The expression \( 2^n - 1 \) can be factored when \( n \) is composite as follows:
\[ 2^{ab} - 1 = (2^a - 1)(2^a + 1)(2^{a(b-1)} + 2^{a(b-2)} + ... + 1) \]
For \( n = 55 \), we can factor \( 55 \) as \( 5 \times 11 \):
\[ 2^{55} - 1 = (2^{5} - 1)(2^{5 \cdot 10} + 2^{5 \cdot 9} + \ldots + 1) \]
Calculating \( 2^5 - 1 \):
\[ 2^5 = 32 \implies 2^5 - 1 = 32 - 1 = 31 \]
Next, let's verify if \( 31 \) is a prime number — it is.
We can also find another factor of \( 2^{55} - 1 \) by noting that \( 2^{55} - 1 \) can be expressed via the factorization for \( n = 11 \):
\[ 2^{11} - 1 = (2^5 - 1)(2^5 + 1) = 31 \cdot 33 \] Where \[ 2^5 + 1 = 32 + 1 = 33 \] Lastly, \( 33 \) can be factored further into \( 3 \times 11 \).
Now, using this, we can conclude:
- \( 2^{55} - 1 = (2^5 - 1)(2^{50} + 2^{45} + \ldots + 1) \)
- Thus, \( 2^{55} - 1 \) is divisible by both \( 31 \) and \( 33 \).
A factor of \( 2^{55} - 1 \) is \( 3 \) since \( 33 = 3 \times 11 \).
So the answer is: \( 2^{55} - 1 \) is not a Mersenne prime, and one of its factors is \( 3 \).