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Difference between revisions of "Mersenne number"
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| − | A '''Mersenne number''' is a number of the form <math>2^n{-}1</math> where <math>n</math> is a non-negative [[ | + | A '''Mersenne number''' is a number of the form <math>2^n{-}1</math> where <math>n</math> is a non-negative [[integer]]. |
| − | When this number is [[ | + | When this number is [[prime]], it is called a [[Mersenne prime]], otherwise it is a [[composite number]]. |
| − | The number of | + | The number of [[digit]]s of a Mersenne number <math>2^n{-}1</math> can be calculated by <math>\lfloor{n*log(2)}\rfloor+1</math>, where <math>\lfloor\rfloor</math> is the floor function (the equivalent to rounding down). |
==Properties of Mersenne numbers== | ==Properties of Mersenne numbers== | ||
Mersenne numbers share several properties: | Mersenne numbers share several properties: | ||
| − | *''M<sub>n</sub>'' is a sum of binomial coefficients: <math> M_n = \sum_{i=0}^{n} {n \choose i} - 1 </math>. | + | *''M<sub>n</sub>'' is a sum of binomial coefficients: <math> M_n = \sum_{i=0}^{n} {n \choose i} - 1</math>. |
| − | + | *If ''a'' is a divisor of ''M<sub>q</sub>'' (''q'' prime) then ''a'' has the following properties: <math>a \equiv 1 \pmod{2q}</math> and: <math>a \equiv \pm 1 \pmod{8}</math>. | |
| − | *If | + | *A theorem from [[Leonhard Euler|Euler]] about numbers of the form ''1+6k'' shows that ''M<sub>q</sub>'' (q prime) is a prime if and only if there exists only one pair <math>(x,y)</math> such that: <math>M_q = {(2x)}^2 + 3{(3y)}^2</math> with <math>q \geq 5</math>. More recently, Bas Jansen has studied <math>M_q = x^2 + dy^2</math> for ''d=0 ... 48'' and has provided a new (and clearer) proof for case ''d=3''. |
| − | + | *Let <math>q = 3 \pmod{4}</math> be a prime. <math>2q+1</math> is also a prime if and only if <math>2q+1</math> divides ''M<sub>q</sub>''. | |
| − | *A theorem from [[Leonhard Euler|Euler]] about numbers of the form ''1+6k'' shows that ''M<sub>q</sub>'' (q prime) is a prime if and only if there exists only one pair <math>(x,y)</math> such that: <math>M_q = {(2x)}^2 + 3{(3y)}^2</math> with <math>q \geq 5</math>. More recently, Bas Jansen has studied <math>M_q = x^2 + dy^2</math> for ''d=0 ... 48'' and has provided a new (and clearer) proof for case d=3. | + | *[[Reix]] has recently found that prime and composite Mersenne numbers ''M<sub>q</sub>'' (q prime > 3) can be written as: <math>M_q = {(8x)}^2 - {(3qy)}^2 = {(1+Sq)}^2 - {(Dq)}^2 </math>. Obviously, if there exists only one pair (x,y), then ''M<sub>q</sub>'' is prime. |
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| − | *Let <math>q = 3 \pmod{4}</math> be a prime. <math>2q+1</math> is also a prime if and only if <math>2q+1</math> divides < | ||
| − | |||
| − | *[[Reix]] has recently found that prime and composite Mersenne numbers < | ||
| − | |||
*[[Ramanujan]] has showed that the equation: <math>M_q = 6+x^2</math> has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite). | *[[Ramanujan]] has showed that the equation: <math>M_q = 6+x^2</math> has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite). | ||
| − | |||
*Any mersenne number is a binary [[repunit]] (in [[base]] 2, they consist of only ones). | *Any mersenne number is a binary [[repunit]] (in [[base]] 2, they consist of only ones). | ||
| − | |||
*If the exponent ''n'' is composite, the Mersenne number must be composite as well. | *If the exponent ''n'' is composite, the Mersenne number must be composite as well. | ||
==External links== | ==External links== | ||
| − | *[ | + | *[[Wikipedia:Mersenne_prime|Wikipedia]] |
*[http://mathworld.wolfram.com/MersenneNumber.html MathWorld] | *[http://mathworld.wolfram.com/MersenneNumber.html MathWorld] | ||
[[Category:Numbers]] | [[Category:Numbers]] | ||
Revision as of 09:58, 5 February 2019
A Mersenne number is a number of the form [math]\displaystyle{ 2^n{-}1 }[/math] where [math]\displaystyle{ n }[/math] is a non-negative integer.
When this number is prime, it is called a Mersenne prime, otherwise it is a composite number.
The number of digits of a Mersenne number [math]\displaystyle{ 2^n{-}1 }[/math] can be calculated by [math]\displaystyle{ \lfloor{n*log(2)}\rfloor+1 }[/math], where [math]\displaystyle{ \lfloor\rfloor }[/math] is the floor function (the equivalent to rounding down).
Properties of Mersenne numbers
Mersenne numbers share several properties:
- Mn is a sum of binomial coefficients: [math]\displaystyle{ M_n = \sum_{i=0}^{n} {n \choose i} - 1 }[/math].
- If a is a divisor of Mq (q prime) then a has the following properties: [math]\displaystyle{ a \equiv 1 \pmod{2q} }[/math] and: [math]\displaystyle{ a \equiv \pm 1 \pmod{8} }[/math].
- A theorem from Euler about numbers of the form 1+6k shows that Mq (q prime) is a prime if and only if there exists only one pair [math]\displaystyle{ (x,y) }[/math] such that: [math]\displaystyle{ M_q = {(2x)}^2 + 3{(3y)}^2 }[/math] with [math]\displaystyle{ q \geq 5 }[/math]. More recently, Bas Jansen has studied [math]\displaystyle{ M_q = x^2 + dy^2 }[/math] for d=0 ... 48 and has provided a new (and clearer) proof for case d=3.
- Let [math]\displaystyle{ q = 3 \pmod{4} }[/math] be a prime. [math]\displaystyle{ 2q+1 }[/math] is also a prime if and only if [math]\displaystyle{ 2q+1 }[/math] divides Mq.
- Reix has recently found that prime and composite Mersenne numbers Mq (q prime > 3) can be written as: [math]\displaystyle{ M_q = {(8x)}^2 - {(3qy)}^2 = {(1+Sq)}^2 - {(Dq)}^2 }[/math]. Obviously, if there exists only one pair (x,y), then Mq is prime.
- Ramanujan has showed that the equation: [math]\displaystyle{ M_q = 6+x^2 }[/math] has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite).
- Any mersenne number is a binary repunit (in base 2, they consist of only ones).
- If the exponent n is composite, the Mersenne number must be composite as well.
