Mersenne primes are prime numbers that can be written in the form 2^n-1 where n is a whole number. For example , 3 can be written as 2^2-1 a) Write a sentence to explain why 2^4-1 is not a Mersenne prime. b) Which two of the following numbers are Mersenne primes? 511 1023

## Step1: A Mersenne prime number is a prime number that can be written in the form , where n represents a whole number. Moreover, not all formulas in the form are prime numbers. We know that to be prime is to have exactly two unique factors - that number and 1. ## Step2: ### For part a)To find out if is a Prime, we'll have a look at its factors. When we calculate , it provides 15.Solve for which is equal to `16 -1` offering us `15`.Identifiable factors of 15 are: 1, 3, 5, and 15.Therefore, or 15 is not a Prime number - and thus temperamentally - not a Mersenne Prime too, as it does not meet the prime number criterion of defining only unique factors of 1 and itself.## Step3: ### For part b) For the numbers provided, they first have to be identified as Prime (having only unique factors of 1 and itself).And then if true, they must calm down or console the Mersenne industry of .1. 17 is a prime, as 17 only factors are 1 and 172. 31 can be written in the form of , and its only factors are 1 and itself.3. 127 can be written in the form of , it satisfies the form of Mersenne prime and its only factors are 1 and itself.4. 129 is not a prime as it has multiple factors5. 511 is not a prime as it has multiple factors6. 1023 is not a prime as it had multiple factorsThe numbers amongst these options that satisfy the criteria for a Mersenne Prime number are 31 and 127.