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

Question

Option #1: 29−1


Option #2: 211−1


Option #3: 215−1


(1 point)

Answer 1

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

Let's analyze the options provided:

1. Option #1: \( 2^9 - 1 \)

   • Here, \( p = 9 \) which is not a prime number.
   • Therefore, this expression does not yield a Mersenne prime.

2. Option #2: \( 2^{11} - 1 \)

   • Here, \( p = 11 \) which is a prime number.
   • Therefore, this expression could yield a Mersenne prime. In fact, \( 2^{11} - 1 = 2047 \) is not prime as it equals \( 23 \times 89 \), but it is in the correct form.

3. Option #3: \( 2^{15} - 1 \)

   • Here, \( p = 15 \) which is not a prime number.
   • Therefore, this expression does not yield a Mersenne prime.

The only option that fits the Mersenne prime definition is Option #2.

The correct option number is 2.