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!

#HelpMe! #MathProblem #Math

Easy Math Editor

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Comments

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

p-th Mersenne number where p is a prime is never a prime power.
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 - 8 years, 3 months ago

Found Fact 1 here.

Zi Song Yeoh - 8 years, 3 months ago

Thanks!

Tan Li Xuan - 8 years, 3 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.

Daniel Liu - 8 years, 3 months ago

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

Tan Li Xuan - 8 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$