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Mersenne number

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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] (see floor function).

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.

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Prime numbers