# 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
5 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.

- 5 years, 8 months ago

Found Fact 1 here.

- 5 years, 8 months ago

Thanks!

- 5 years, 8 months ago

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.

- 5 years, 8 months ago

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

- 5 years, 8 months ago