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Mersenne primes

Hi everyone,I have some questions about Mersenne primes. ( Mersenne primes are prime numbers in the form of \( 2^{p}-1 \) where \( p \) is an integer )Please help.Thanks! The questions are:

1.Double Mersenne primes are in the form of \( 2^{2^{p}-1}-1 \).If so,then what is the form of a triple Mersenne prime,if they exist?
2.Are there any double Mersenne primes other than the four known?

3.Is there any known formula to determine Mersenne primes?

4.Is there a limit to the number of Mersenne primes that can be calculated on a computer with a fixed processing power (say 1 Ghz )?(this is more of a computing question....)

Thanks again!

Note by Tan Li Xuan
4 years, 8 months ago

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I was finding information on Mersenne Numbers and found some facts:

  1. p-th Mersenne number where p is a prime is never a prime power.

  2. If \(2^p - 1\) is a Mersenne prime, then p is a prime.

Note : This does not mean that if p is a prime, then \(2^p - 1\) is a prime.

Zi Song Yeoh - 4 years, 8 months ago

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Found Fact 1 here.

Zi Song Yeoh - 4 years, 8 months ago

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Thanks!

Tan Li Xuan - 4 years, 8 months ago

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From http://primes.utm.edu/mersenne/: Lucas-Lehmer Test: For \(p\) an odd prime, the Mersenne number \(2p-1\) is prime if and only if \(2p-1\) divides \(S(p-1)\) where \(S(n+1) = S(n)2-2\), and \(S(1) = 4\). This is to answer your question "Is there any known formula to determine Mersenne primes?" Hope this helped.

Daniel Liu - 4 years, 8 months ago

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Thanks Daniel,that is really helpful! :) But Mersenne primes are in the form of \( 2^{p}-1 \),not \( 2p-1 \).

Tan Li Xuan - 4 years, 8 months ago

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