Which of the following statements must be true about n in the expression for a Mersenne prime, 2n−1

Question

? Enter the number of the correct statement.

Statement #1: It must be an even number.

Statement #2: It must be an odd composite number.

Statement #3: It must be a prime number.

(1 point)

Answer 1

To determine which statements about \( n \) in the expression for a Mersenne prime \( 2^n - 1 \) must be true, let’s evaluate each statement:

1. Statement #1: It must be an even number.

   • False. If \( n \) is even, \( 2^n - 1 \) can be expressed as \( (2^{n/2} - 1)(2^{n/2} + 1) \), which means it is not prime unless \( n = 2 \). Therefore, \( n \) should be odd for \( 2^n - 1 \) to potentially be a prime.

2. Statement #2: It must be an odd composite number.

   • False. For a Mersenne prime, \( n \) must be a prime number, not a composite number.

3. Statement #3: It must be a prime number.

   • True. For \( 2^n - 1 \) to be a Mersenne prime, \( n \) needs to be a prime number. If \( n \) is not prime (composite), then \( 2^n - 1 \) will not be a prime.

Therefore, the correct statement is:

Statement #3: It must be a prime number.

Final Answer: 3