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 ${\displaystyle \lfloor n\ast log(2)\rfloor +1}$ (see floor function).

Properties of Mersenne numbers

Mersenne numbers share several properties:

• M_n is a sum of binomial coefficients: ${\displaystyle M_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})-1}$.
• If a is a divisor of M_q (q prime) then a has the following properties: ${\displaystyle a\equiv 1(\mathrm{mod}2q)}$ and: ${\displaystyle a\equiv \pm 1(\mathrm{mod}8)}$.
• 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 ${\displaystyle (x,y)}$ such that: ${\displaystyle M_q={(2x)}^2+3{(3y)}^2}$ with ${\displaystyle q\ge 5}$. More recently, Bas Jansen has studied ${\displaystyle M_q=x^2+d{y}^2}$ for d=0 ... 48 and has provided a new (and clearer) proof for case d=3.
• Let ${\displaystyle q=3(\mathrm{mod}4)}$ be a prime. ${\displaystyle 2q+1}$ is also a prime if and only if ${\displaystyle 2q+1}$ divides M_q.
• Reix has recently found that prime and composite Mersenne numbers M_q (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 M_q 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.

External links

• Mersenne prime
• Mersenne number