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# Mersenne number

A Mersenne number is a number of the form $\displaystyle{ 2^n{-}1 }$ where $\displaystyle{ n }$ 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 $\displaystyle{ 2^n{-}1 }$ can be calculated by $\displaystyle{ \lfloor{n*log(2)}\rfloor+1 }$ (see floor function).

## Properties of Mersenne numbers

Mersenne numbers share several properties:

• Mn is a sum of binomial coefficients: $\displaystyle{ M_n = \sum_{i=0}^{n} {n \choose i} - 1 }$.
• If a is a divisor of Mq (q prime) then a has the following properties: $\displaystyle{ a \equiv 1 \pmod{2q} }$ and: $\displaystyle{ a \equiv \pm 1 \pmod{8} }$.
• 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 $\displaystyle{ (x,y) }$ such that: $\displaystyle{ M_q = {(2x)}^2 + 3{(3y)}^2 }$ with $\displaystyle{ q \geq 5 }$. More recently, Bas Jansen has studied $\displaystyle{ M_q = x^2 + dy^2 }$ for d=0 ... 48 and has provided a new (and clearer) proof for case d=3.
• Let $\displaystyle{ q = 3 \pmod{4} }$ be a prime. $\displaystyle{ 2q+1 }$ is also a prime if and only if $\displaystyle{ 2q+1 }$ divides Mq.
• Reix has recently found that prime and composite Mersenne numbers Mq (q prime > 3) can be written as: $\displaystyle{ M_q = {(8x)}^2 - {(3qy)}^2 = {(1+Sq)}^2 - {(Dq)}^2 }$. Obviously, if there exists only one pair (x,y), then Mq is prime.
• Ramanujan has showed that the equation: $\displaystyle{ M_q = 6+x^2 }$ 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.