Question

Determine whether 2^55 -1 is a Mersenne prime if not find the factor of 2^55-1

A. No 2^55-1 is not a Mersenne prime a factor of 2^55 -1 is 2^10 +2^9 + 2^8 +2^7+2^6+2^5+2^4+2^3+2^2+2+1=2,047

B. No 2^55-1 is not a prime a factor of 2^55-1 is -1 is 2^5 +2^4 +2^3+2^2+2+1=63
C. No 2^55-1 is not a prime a factor of 2^55-1 is 2^22-1
D. Yes 2^55 -1 is a Mersenne prime

Answer 1

Are you sure

Answer 2

To determine whether \( 2^{55} - 1 \) is a Mersenne prime, we first need to recall that a Mersenne prime has the form \( 2^p - 1 \), where \( p \) is a prime number. In this case, \( p = 55 \), and since 55 is not a prime number (it factors as \( 5 \times 11 \)), \( 2^{55} - 1 \) cannot be a Mersenne prime.

Next, we can find the factors of \( 2^{55} - 1 \). The expression \( 2^{n} - 1 \) can be factored using the difference of squares and other algebraic identities. Specifically, we can factor \( 2^{55} - 1 \) as follows:

\[
2^{55} - 1 = (2^{5} - 1)(2^{50} + 2^{45} + 2^{40} + 2^{35} + 2^{30} + 2^{25} + 2^{20} + 2^{15} + 2^{10} + 2^{5} + 1)
\]

Now, calculating \( 2^{5} - 1 \):

\[
2^5 - 1 = 32 - 1 = 31
\]

So we have:

\[
2^{55} - 1 = 31 \times (36028797018963968 + ...)
\]

We can also note that \( 2^{n} - 1 \) can be factored into smaller terms. Another way to express the factors related to \( n = 55 \) would be:

\[
2^{55} - 1 = (2^{11} - 1)(2^{44} + 2^{33} + 2^{22} + 2^{11} + 1)
\]

Calculating \( 2^{11} - 1 \):

\[
2^{11} - 1 = 2048 - 1 = 2047
\]

Thus, \( 2^{55} - 1 \) can be factored further into:

\[
2^{55} - 1 = 2047 \times (2^{44} + 2^{33} + 2^{22} + 2^{11} + 1)
\]

Now among the options provided:

- Option A states a factor as \( 2047 \) (which is \( 2^{11} - 1 \)).
- Option B states a factor as \( 63 \) which is incorrect since \( 2^6 - 1 = 63 \) does not directly relate here.
- Option C states \( 2^{22} - 1 \) which is actually true.
- Option D claims that \( 2^{55}-1 \) is a Mersenne prime, which is false.

Given the options, the correct answer is:

**A. No, \( 2^{55} - 1 \) is not a Mersenne prime. A factor of \( 2^{55} - 1 \) is \( 2047.**

Answer 3

Let's go through the reasoning step by step to ensure clarity and correctness regarding whether \( 2^{55} - 1 \) is a Mersenne prime, and we'll confirm the factorization details.

1. **Definition of Mersenne Prime**:
A Mersenne prime is of the form \( 2^p - 1 \), where \( p \) is a prime number.

2. **Evaluating \( p \)**:
Here, \( p = 55 \). Since 55 is not a prime number (it factors into \( 5 \times 11 \)), this means \( 2^{55} - 1 \) cannot be a Mersenne prime.

3. **Factoring \( 2^{55} - 1 \)**:
We can use the identity for factoring \( 2^n - 1 \):

\[
2^{n} - 1 = (2^{k} - 1)(2^{n-k} + 2^{n-2k} + \ldots + 2^{k} + 1)
\]

For \( n = 55 \):

- It factors as:
\[
2^{55} - 1 = (2^{5} - 1)(2^{50} + 2^{45} + \cdots + 1)
\]

- We can also further break down:
\[
2^{5} - 1 = 31
\]
- So one factor is \( 31 \).

- There’s also a factorization from:
\[
2^{55} - 1 = (2^{11} - 1)(2^{44} + 2^{33} + 2^{22} + 2^{11} + 1)
\]
where:
\[
2^{11} - 1 = 2047
\]

4. **Finding Factors**:
- Another factor is \( 2^{22} - 1 \), since it can also factor \( 2^{55} - 1 \) from repeating the above rules.

Based on the analysis:
- \( 2^{55} - 1 \) is **not** a Mersenne prime.
- It can be factored into \( 31 \), \( 2047 \), etc.

### Conclusion:
From the options:

- **A** correctly identifies that \( 2^{55} - 1 \) is not a Mersenne prime and \( 2047 \) is a valid factor.
- **B** incorrectly claims a factor of \( 63 \).
- **C** mentions \( 2^{22} - 1 \), which is true as a factor.
- **D** asserts it is a Mersenne prime, which is false.

So I can confirm the correctness of option **A**: \( 2^{55} - 1 \) is **not** a Mersenne prime, and \( 2047 \) is indeed a factor. Thank you for your patience as we clarified this.