Question
s 2^(87)-1 a Mersenne prime? If not, find one of its factors
A.It is not possible to determine
B.No
C. Yes
A.It is not possible to determine
B.No
C. Yes
Answers
GPT-4o mini
To determine whether \( 2^{87} - 1 \) is a Mersenne prime, we need to check if it is prime. A Mersenne prime is in the form \( 2^p - 1 \), where \( p \) is itself prime.
First, let's check if 87 is a prime number. The number 87 can be factored as \( 3 \times 29 \). Since 87 is not prime (it has divisors other than 1 and itself), \( 2^{87} - 1 \) cannot be a Mersenne prime.
Thus, \( 2^{87} - 1 \) is not a prime number. We can also find one of its factors. Using the property that \( 2^n - 1 \) can be factored for composite \( n \):
\[
2^{87} - 1 = (2^{29} - 1)(2^{58} + 2^{29} + 1)
\]
Here, \( 2^{29} - 1 \) is one of its factors.
Therefore, the answer to whether \( 2^{87} - 1 \) is a Mersenne prime is **B. No**.
First, let's check if 87 is a prime number. The number 87 can be factored as \( 3 \times 29 \). Since 87 is not prime (it has divisors other than 1 and itself), \( 2^{87} - 1 \) cannot be a Mersenne prime.
Thus, \( 2^{87} - 1 \) is not a prime number. We can also find one of its factors. Using the property that \( 2^n - 1 \) can be factored for composite \( n \):
\[
2^{87} - 1 = (2^{29} - 1)(2^{58} + 2^{29} + 1)
\]
Here, \( 2^{29} - 1 \) is one of its factors.
Therefore, the answer to whether \( 2^{87} - 1 \) is a Mersenne prime is **B. No**.