Question

Use Exercise 41 to determine whether

M _13​ =2^ 13 −1=8191 and M _23 =2 ^23 −1=8,388,607 are prime.

a) Both Prime
b) M _13Prime,
M _23Not Prime
c) M _13Not Prime,
M _23Prime
d) Both Not Prime

Answer 1

To determine whether M_13 = 2^13 - 1 = 8191 and M_23 = 2^23 - 1 = 8,388,607 are prime, we can check if they are divisible by any prime numbers less than or equal to the square root of the numbers.

Step-by-step explanation:

To determine whether M13 = 213 - 1 = 8191 and M23 = 223 - 1 = 8,388,607 are prime, we need to check for any factors other than 1 and the number itself. One way to do this is by testing if any prime numbers are less than or equal to the square root of the number and dividing it evenly. These numbers are known as Mersenne primes, which are prime numbers of the form Mn = 2n − 1 where n is itself a prime number.

For M13 = 8191, we can check if it is divisible by any prime numbers less than or equal to √8191 ≈ 90.46. Testing prime numbers up to 90, we find that 8191 is not divisible by any of them, which indicates that it is prime.

Similarly, for M23 = 8388607, we can check if it is divisible by any prime numbers less than or equal to √8388607 ≈ 2898.5. Testing prime numbers up to 2898, we find that 8388607 is not divisible by any of them, which also indicates that it is prime.