Question

Determine whether the following statements are true or false: 1) If n is a prime number, then the Mersenne number 2ⁿ -1 is also a prime number. A) True

B) False 2) If n is a prime number, then the Mersenne number 2ⁿ -1 is also a composite number. A) True
B) False

Answer 1

The statement 1 is false because not all Mersenne numbers with prime exponents are necessarily prime numbers. The statement 2 is true because there are prime values of n that result in composite Mersenne numbers.

Step-by-step explanation:

The statement 1) If n is a prime number, then the Mersenne number 2ⁿ -1 is also a prime number, is false. This is because not all Mersenne numbers with prime exponents are necessarily prime numbers. For example, 2² - 1 = 3, which is prime, but 2⁷ - 1 = 127, which is also prime. However, 2¹¹ - 1 = 2047 = 23 x 89, which is composite.

The statement 2) If n is a prime number, then the Mersenne number 2ⁿ - 1 is also a composite number, is true. As shown in the previous example, there are prime values of n that result in composite Mersenne numbers.