Since monk Marine Mersenne studied the primality of in 1644, Mersenne primes, i.e., ( : prime), have been developed by numerous researchers, such as Euler, Lucas, Pervouchine, Cole, and Powers, and in recent years, by GIMPS (Great Internet Mersenne Prime Search).
If is prime, then n is also prime, because if , ( ), then (ab digits in binary) can be divided by (a digits in binary). However, the converse is not true, for example, .
In addition, it is well known that all even perfect numbers (odd perfect numbers are unknown) are generated by , if and only if is prime.
The current Mersenne prime numbers are denoted by , for
. The most recent Mersenne prime number is (22338618 digits), which was developed in January 2016.
2. Bounds for the Sum of Reciprocals of Mersenne Primes
We begin by defining the notation. We define
: k-th number in S.
We can effectively calculate , as rapidly converges to 0.
For example, if we consider , we obtain
which provides the value of up to 17 decimal digits. If we con-
sider , we can precisely calculate the sum of reciprocals of Mersenne primes up to 156 decimal digits, which is given by
According to the Goldbach-Euler theorem  ,
where is the set of perfect powers of
Theorem 2. The sum of reciprocals of Mersenne prime numbers is larger than that of where , namely,
Proof. It holds that
Considering that , it follows from Goldbach-Euler theorem that
Figure 1. Relationship between and .
We should note that the sum of reciprocals of prime numbers appears to converge numerically; however, it is infinite, which is proved in, e.g., Hardy and Wright  . Therefore, Mersenne primes are considerably sparse subsequences of prime numbers.
In the case of twin primes, the value of the sum of reciprocals of twin primes is shown to be bounded above by Brun  and is estimated as
which is known as Brun’s constant (however, it is an estimation). Even though the problem that whether twin primes are infinite is still unsolved, Zhang  presented an important result, which states that a constant exists between two successive primes that are infinite. If we can lower the upper bound by 4, the twin prime conjecture will be solved. The Polymath8 project has reduced the upper bound by four digits.
In addition, the problem that whether Mersenne primes are infinite is still unresolved. Figure 1 shows the relationship between and .
Denote the number of Mersenne primes that do not exceed x. Then, as for , it seems from Figure 1 that
numerically, where c is a constant. In other words, the number of Mersenne primes tends to increase by a constant per digit.
 Brun, V. (1919) La série
1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+··· les dénominateurs sont nombres premiers jumeaux est convergente où finie. Bulletin des Sciences Mathématiques, 43, 124-128. (In French)