Difference between revisions of "Mersenne prime"
m (Changed "The largest prime is ...." to "The largest known prime is ...." I'm fairly certain that 2^43112609-1 isn't the largest prime out there.) |
Scrabbler94 (talk | contribs) (update largest known Mersenne prime; connection with even perfect numbers) |
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| − | A Mersenne [[prime]] is a prime that is in the form of <math>2^n-1</math>. | + | A Mersenne [[Prime number|prime]] is a prime that is in the form of <math>2^n-1</math>, where <math>n</math> is an integer. It is named after Marin Mersenne. |
| − | These are some of the largest primes known to man due to one main factor: There is an integer bit value set to that, so that the largest number with a certain amount of bits is a form of <math>2^n-1</math> | + | These are some of the largest primes known to man due to one main factor: There is an integer bit value set to that, so that the largest number with a certain amount of bits is a form of <math>2^n-1</math>. |
For example: The amount of numbers on a 32 bit computer is <math>2^{32}</math>. Then, divide by 2, as there are positive, and negative values. Then subtract one, as zero is one of them, so the largest number on a 32 bit computer is 2,147,483,647. (Not necessarily the largest number displayed, to achieve a higher number, a computer could use a base system other than 2.) | For example: The amount of numbers on a 32 bit computer is <math>2^{32}</math>. Then, divide by 2, as there are positive, and negative values. Then subtract one, as zero is one of them, so the largest number on a 32 bit computer is 2,147,483,647. (Not necessarily the largest number displayed, to achieve a higher number, a computer could use a base system other than 2.) | ||
| − | + | As of January 2018, the largest known prime is <math>2^{77,232,917}-1</math>, a Mersenne prime which contains 23,249,425 digits. | |
| + | |||
| + | ==Connection with Even Perfect Numbers== | ||
| + | All even [[perfect number|perfect numbers]] are of the form <math>\frac{p(p+1)}{2}</math> where <math>p = 2^k-1</math> is a Mersenne prime, which was proven by Euler in the 18th century. | ||
Revision as of 16:07, 22 February 2018
A Mersenne prime is a prime that is in the form of 
, where 
is an integer. It is named after Marin Mersenne.
These are some of the largest primes known to man due to one main factor: There is an integer bit value set to that, so that the largest number with a certain amount of bits is a form of .
For example: The amount of numbers on a 32 bit computer is 
. Then, divide by 2, as there are positive, and negative values. Then subtract one, as zero is one of them, so the largest number on a 32 bit computer is 2,147,483,647. (Not necessarily the largest number displayed, to achieve a higher number, a computer could use a base system other than 2.)
As of January 2018, the largest known prime is 
, a Mersenne prime which contains 23,249,425 digits.
Connection with Even Perfect Numbers
All even perfect numbers are of the form 
where 
is a Mersenne prime, which was proven by Euler in the 18th century.
