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

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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

Answer 1

a) \(2^{4} - 1\) is not a Mersenne prime, as partly because \(2^{4} - 1 = 15\) which has more than 2 unique factors (1, 3, 5, and 15) b) The two Mersenne primes from the list provided are 31 and 127.

Answer 2

## Step1: A Mersenne prime number is a prime number that can be written in the form \(2^{n}-1\), where n represents a whole number. Moreover, not all formulas in the \(2^n -1\) 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 \(2^{4}-1\) is a Prime, we'll have a look at its factors. When we calculate \(2^{4}-1\), it provides 15. Solve for \(2^{4}-1\) which is equal to `16 -1` offering us `15`. Identifiable factors of 15 are: 1, 3, 5, and 15. Therefore, \(2^{4}-1\) 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 \(2^n -1\). 1. 17 is a prime, as 17 only factors are 1 and 17 2. 31 can be written in the form of \(2^5 -1\), and its only factors are 1 and itself. 3. 127 can be written in the form of \(2^7 - 1\), 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 factors 5. 511 is not a prime as it has multiple factors 6. 1023 is not a prime as it had multiple factors The numbers amongst these options that satisfy the criteria for a Mersenne Prime number are 31 and 127.

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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

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