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Mersenne number
A Mersenne number is a number of the form [math]2^n{}1[/math] where [math]n[/math] is a nonnegative 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]2^n{}1[/math] can be calculated by [math]\lfloor{n*log(2)}\rfloor+1[/math] (see floor function).
Properties of Mersenne numbers
Mersenne numbers share several properties:
 M_{n} 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_{q} (q prime) then a has the following properties: [math]a \equiv 1 \pmod{2q}[/math] and: [math]a \equiv \pm 1 \pmod{8}[/math].
 A theorem from Euler about numbers of the form 1+6k shows that M_{q} (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_{q}.
 Reix has recently found that prime and composite Mersenne numbers M_{q} (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_{q} is prime.
 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).
 If the exponent n is composite, the Mersenne number must be composite as well.
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