Which of the following expressions could, by definition, yield a Mersenne prime number? Enter the

Question

number of the correct option.
Option #1:2^9 - 1
Option #2:2^11 -1
Option #3:2^15 -1

Answer 1

A Mersenne prime is defined as a prime number that can be expressed in the form \(2^p - 1\), where \(p\) is also a prime number.

Let's evaluate the provided options:

1. Option #1: \(2^9 - 1\)
   Here, \(p = 9\). Since 9 is not a prime number, \(2^9 - 1\) cannot be a Mersenne prime.

2. Option #2: \(2^{11} - 1\)
   Here, \(p = 11\). Since 11 is a prime number, \(2^{11} - 1\) could be a Mersenne prime. In fact, \(2^{11} - 1 = 2047\), which can be factored as \(23 \times 89\), so it is not prime.

3. Option #3: \(2^{15} - 1\)
   Here, \(p = 15\). Since 15 is not a prime number, \(2^{15} - 1\) cannot be a Mersenne prime.

Based on the evaluation, the only expression that can theoretically yield a Mersenne prime in form is Option #2, but it does not yield a prime itself.

Thus, the number of the correct option that could yield a Mersenne prime number is:

2