Question

Determine whether each of these integers is prime, verifying some of Mersenne's claims.

a) 2^7 − 1
b) 2^9 − 1
c) 2^11 − 1
d) 2^13 − 1

Answer 1

Of the Mersenne numbers given, 2^7 - 1 and 2^13 - 1 are prime, while 2^9 - 1 and 2^11 - 1 are not prime.

Step-by-step explanation:

Determining whether an integer is prime involves checking if it has any divisors other than 1 and itself. For the Mersenne numbers given, which are of the form 2n − 1 with n being a prime number, we check the integers for primality:

• a) 27 − 1 = 127, which is prime.
• b) 29 − 1 = 511, which is not prime (it is 7 × 73).
• c) 211 − 1 = 2047, which is not prime (it is 23 × 89).
• d) 213 − 1 = 8191, which is prime.

Therefore, the integers from a) 27 − 1 and d) 213 − 1 are prime, whereas b) 29 − 1 and c) 211 − 1 are not prime.