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

Question

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

GPT-4o mini · Answer

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

Are you sure

GPT-4o mini · Answer

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.

Definition of Mersenne Prime: A Mersenne prime is of the form $ 2^p - 1 $, where $ p $ is a prime number.
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.
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: \[ 2^{11} - 1 = 2047 \]
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.