What Is Mersenne Number just wondering

[AlanDS 0]

I have heard this (mersenne number) many times, but I have never come across it in my college. Anyone here knows whats it?

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Edited May 22, 2006 by BuffaloHELP (see edit history)

[AceGirl 0]

I would love to know what that is also. I saw a thread in here somewhere mentioning it but i dont know what it was. If anyone does know please share with us eager learners!

[Outsider 0]

Well, I too havent heard of it. But here's what Wikipedia says:

 

Mersenne prime

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In mathematics, a Mersenne prime is a prime number that is one less than a prime power of two. For example, 31 (a prime number) = 32 − 1 = 25 − 1, and 5 also a prime number, so 31 is a Mersenne prime; so is 7 = 8 − 1 = 23 − 1. On the other hand, 2047 = 2048 − 1 = 211 − 1, for example, is not a prime, because although 11 is a prime (making it a candidate for being a Mersenne prime), 2047 is not prime (it is divisible by 89 & 23). Throughout modern times, the largest known prime number has very often been a Mersenne prime.

 

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a prime power of two; hence,

 

Mn = 2n − 1.

 

(most sources restrict the term Mersenne number to where n is prime as all Mersenne primes must be of this form as seen below)

 

Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exists (any that do have to belong to a significant number of special forms; see perfect number for more details).

 

It is currently unknown whether there is an infinite number of Mersenne primes.

Great amount of search is going on in this matter. Here is what http://www.mersenne.org/ says:

 

What are Mersenne primes and why do we search for them?

Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime of the form 2P-1. The first Mersenne primes are 3, 7, 31, 127 (corresponding to P = 2, 3, 5, 7). There are only 43 known Mersenne primes.

 

GIMPS, the Great Internet Mersenne Prime Search, was formed in January 1996 to discover new world-record-size Mersenne primes. GIMPS harnesses the power of thousands of small computers like yours to search for these "needles in a haystack".

 

Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. Of course, there are many other reasons.

[miCRoSCoPiC^eaRthLinG 0]

A Mersenne number is actually anumber which can expressed in the form 2^n-1 (one less than a power of two). A Mersenne number which is a Prime is known as a Mersenne Prime.

 

There's an organization called GIMPS (Great Internet Mersenne Prime Search) - who keeps trying to find the greatest mersenne prime using hi-tech supercomputing clusters.

 

Electronic Frontier Foundation has even got a nice sum up as prize money ($100,000) for whoever discovers the first 10 million digit Mersenne Prime.

 

On December 15, 2005, Dr. Curtis Cooper and Dr. Steven Boone, professors at Central Missouri State University, discovered the 43rd Mersenne Prime, 230,402,457-1. The CMSU team is the most prolific contributor to the GIMPS project. The discovery is the largest known prime number.

 

The new prime is 9,152,052 digits long

 

More resources on this:

1. http://www.mersenne.org/

2. http://primes.utm.edu/mersenne/

3. http://primes.utm.edu/largest.html

 

 

This is actually quite an old topic at Asta - I'd made a post on this sometime last year and we've had a lively discussion on it. The thread can be found at: What Is: World's Largest Known Prime Number. Check it out.

[AlanDS 0]

Thanks Guys!So 2^(something) - 1 is a mersenne number or Mersenne prime.That post mentioned above was interesting too!

[miCRoSCoPiC^eaRthLinG 0]

Thanks Guys!

 

So 2^(something) - 1 is a mersenne number or Mersenne prime.

 

That post mentioned above was interesting too!

 

Not exactly.. See, any number that can be expressed in the form 2^n-1 is a Mersenne Number.

 

For example:

2^2 - 1 = 3

 

3 here is a Mersenne number.. but so is 15,

 

2^4 - 1 = 15

 

The difference among these is that 3 is a PRIME - since, it's perfectly divisible only by itself and 1. But 15 is NOT a prime.. (since it's divisible by 3 & 5 other than itself and 1)

 

Thus a number which can be expressed as 2^n-1 AND is a prime - is called a Mersenne Prime.