Question
If $2^n-1$ is prime, then must $n$ be prime?
Step 1
Step 1: We start by analyzing the statement: "If \(2^n - 1\) is prime, then must \(n\) be prime?" We need to determine if \(n\) being composite (not prime) can lead to \(2^n - 1\) being prime. Show more…
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Show that if $2^{n}-1$ is prime, then $n$ is prime. [Hint: Use the identity $2^{a b}-1=\left(2^{a}-1\right) \cdot\left(2^{a(b-1)}+2^{a(b-2)}+\cdots+\right.$ $2^{a}+1 ) . ]$
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A number of the form 2^n -1 is called a Mersenne number, and if it happens to be prime it is called a Mersenne prime. Some exceptionally large Mersenne primes have been discovered. Question : If n is greater than or equal to 1 and 2^n -1 is a prime, prove that n is prime. Hint. Start with supposing n is not prime. Then use the following factorization, which holds for any x and any positive m. x^m - 1 = (x - 1) (x^(m-1) + x^(m-2)..... + x^2 + x + 1)
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