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The sum of reciprocals of Mersenne primes converges to 0.51645417894078856533 ··· , which is an example of a probably infinite subset of primes whose sum of reciprocals is finite and can be computed accurately. This value is larger than , wh ere i s t he set of perfect powers of prime numbers.

Since monk Marine Mersenne studied the primality of M n = 2 n − 1 in 1644, Mersenne primes, i.e., M n = 2 n − 1 ( M n : 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 M n is prime, then n is also prime, because if n = a b , ( a , b ≥ 2 ), then M n = 11 ⋯ 1 (ab digits in binary) can be divided by 1 ⋯ 1 (a digits in binary). However, the converse is not true, for example, M 11 = 2047 = 23 × 89 .

In addition, it is well known that all even perfect numbers (odd perfect numbers are unknown) are generated by 2 n − 1 M n , if and only if M n is prime.

The current Mersenne prime numbers are denoted by M n , for

n = 2 , 3 , 5 , 7 , 13 , 17 , 19 , 31 , 61 , 89 , 107 , 127 , 521 , ⋯ . The most recent Mersenne prime number is M 74207281 (22338618 digits), which was developed in January 2016.

We begin by defining the notation. We define

S ≡ { 2 , 3 , 5 , 7 , 13 , 17 , 19 , 31 , 61 , 89 , 107 , 127 , 521 , ⋯ }

i k : k-th number in S.

Theorem 1.

∑ k = 1 N 1 M i k < ∑ M n : primes 1 M n < ∑ k = 1 N 1 M i k + 2 2 − i N + 1 , ∀ i N ∈ S

Proof.

∑ M n : primes 1 M n = ∑ i ∈ S 1 M i = ∑ k = 1 N 1 M i k + ∑ k = N + 1 ∞ 1 M i k < ∑ k = 1 N 1 M i k + ∑ k = N + 1 ∞ 1 2 i k − 1 < ∑ k = 1 N 1 M i k + ∑ k = N + 1 ∞ 1 2 i k − 1 = ∑ k = 1 N 1 M i k + 2 2 − i N + 1

We can effectively calculate ∑ M n : primes 1 M n , as 2 2 − i N + 1 rapidly converges to 0.

For example, if we consider N = 8 , we obtain

0.51645417894078856489 ⋯ < ∑ M n : primes 1 M n < 0.51645417894078856663 ⋯ ,

which provides the value of ∑ M n : primes 1 M n up to 17 decimal digits. If we con-

sider N = 12 , we can precisely calculate the sum of reciprocals of Mersenne primes up to 156 decimal digits, which is given by

0.516454178940788565330487342971522858815968553415419701441931

065273568701440212723499154883293666215374032432110836569575

419140470924868317486037285294641634・・・

According to the Goldbach-Euler theorem [

∑ q ∈ ℙ 1 q − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + ⋯ = 1 ,

where ℙ ≡ { p i | p isaprimenumber ≥ 2 and i ≥ 2 } is the set of perfect powers of

prime numbers.

Theorem 2. The sum of reciprocals of Mersenne prime numbers is larger than that of q − 1 where q ∈ ℙ | p ≥ 3 , namely,

∑ M n : primes 1 M n > ∑ { q ∈ ℙ ∣ p ≥ 3 } 1 q − 1

Proof. It holds that

1 M 2 + 1 M 3 + 1 M 5 ≈ 0.50844854 ⋯ < ∑ M n : primes 1 M n .

Considering that M n : primes ∈ 2 i − 1 ( i ≥ 2 ) , it follows from Goldbach-Euler theorem that

∑ M n : primes 1 M n + ∑ { q ∈ ℙ ∣ p ≥ 3 } 1 q − 1 < ∑ i ≥ 3 1 2 i − 1 + ∑ { q ∈ ℙ ∣ p ≥ 3 } 1 q − 1 = 1.

Hence,

∑ { q ∈ ℙ ∣ p ≥ 3 } 1 q − 1 < 1 2 < ∑ M n : primes 1 M n ,

since 1 2 < ∑ M n : primes 1 M n .

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 [

In the case of twin primes, the value of the sum of reciprocals of twin primes is shown to be bounded above by Brun [

( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + ( 1 17 + 1 19 ) + ⋯ ≈ 1.9021605823 ± 8 × 10 − 10 ,

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 [

In addition, the problem that whether Mersenne primes are infinite is still unresolved.

Denote m ( x ) the number of Mersenne primes that do not exceed x. Then, as m ( x ) = i n for x = M n , it seems from

m ( x ) ∝ log x + c ,

numerically, where c is a constant. In other words, the number of Mersenne primes tends to increase by a constant per digit.

Tanaka, Y. (2017) On the Sum of Reciprocals of Mersenne Primes. American Journal of Computational Mathematics, 7, 145-148. https://doi.org/10.4236/ajcm.2017.72012