Question
s 2^(87)-1 a Mersenne prime? If not, find one of its factors
Answers
GPT-4o mini
To determine whether \( 2^{87} - 1 \) is a Mersenne prime, we first check if \( 87 \) is prime. The number \( 87 \) is not a prime number because it can be factored as \( 87 = 3 \times 29 \).
For a Mersenne number \( 2^p - 1 \) to be prime, \( p \) itself must also be prime. Since \( 87 \) is composite, \( 2^{87} - 1 \) cannot be prime.
Next, we can use the fact that \( 2^{a} - 1 \) can be factored into \( (2^{b} - 1)(2^{b} + 1) \) when \( a \) is even and \( a \) can be expressed as \( ab \). Specifically, we can factor \( 2^{87} - 1 \) as:
\[
2^{87} - 1 = (2^{29} - 1)(2^{29} + 1)(2^{3} - 1)
\]
Since \( 87 = 3 \times 29 \), we can factor \( 2^{87} - 1 \) as follows:
1. Calculate \( 2^{3} - 1 = 8 - 1 = 7 \).
2. Calculate \( 2^{29} - 1 \).
Now, \( 2^{29} - 1 \) is also a number worth checking. The Mersenne prime candidates occur for primes \( p \). The known primes from before include \( p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127 \), and \( 29 \) is known to lead to a prime \( 2^{29} - 1 \).
Calculating \( 2^{29} - 1 \):
\[
2^{29} = 536870912 \Rightarrow 2^{29} - 1 = 536870911
\]
Now let's check for \( 2^{29} + 1 \):
\[
2^{29} + 1 = 536870913
\]
Thus, one of the factors of \( 2^{87} - 1 \) is \( 7 \) from \( 2^{3} - 1 \).
Therefore, \( 2^{87} - 1 \) is not a Mersenne prime, and one of its factors is \( 7 \).
For a Mersenne number \( 2^p - 1 \) to be prime, \( p \) itself must also be prime. Since \( 87 \) is composite, \( 2^{87} - 1 \) cannot be prime.
Next, we can use the fact that \( 2^{a} - 1 \) can be factored into \( (2^{b} - 1)(2^{b} + 1) \) when \( a \) is even and \( a \) can be expressed as \( ab \). Specifically, we can factor \( 2^{87} - 1 \) as:
\[
2^{87} - 1 = (2^{29} - 1)(2^{29} + 1)(2^{3} - 1)
\]
Since \( 87 = 3 \times 29 \), we can factor \( 2^{87} - 1 \) as follows:
1. Calculate \( 2^{3} - 1 = 8 - 1 = 7 \).
2. Calculate \( 2^{29} - 1 \).
Now, \( 2^{29} - 1 \) is also a number worth checking. The Mersenne prime candidates occur for primes \( p \). The known primes from before include \( p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127 \), and \( 29 \) is known to lead to a prime \( 2^{29} - 1 \).
Calculating \( 2^{29} - 1 \):
\[
2^{29} = 536870912 \Rightarrow 2^{29} - 1 = 536870911
\]
Now let's check for \( 2^{29} + 1 \):
\[
2^{29} + 1 = 536870913
\]
Thus, one of the factors of \( 2^{87} - 1 \) is \( 7 \) from \( 2^{3} - 1 \).
Therefore, \( 2^{87} - 1 \) is not a Mersenne prime, and one of its factors is \( 7 \).