# Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

57 questions
0answers
43 views

### Division by Mersenne primes

Mersenne primes are used in Computer Science and Cryptography because they support fast modulo computation. If $p$ is a Mersenne prime, $n \bmod p$ can be computed with just a few add and shift ...
3answers
112 views

1answer
110 views

0answers
89 views

### Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...
1answer
47 views

1answer
122 views

### Could a Mersenne prime divide an odd perfect number?

The relationship between Mersenne primes $2^r-1$ and even perfect numbers $2^{r-1}(2^r-1)$ is well-known (Euclid, Euler). In a video on the web I heard the statement that it is known that a Mersenne ...
1answer
56 views

1answer
95 views

### How many numbers $2^n-k$ are prime?

We are all familiar with the Mersenne primes $$M_n = 2^n-1$$ and we indeed know that there are some $M_n$ that are prime. However, it is still open whether there are infinitly many $M_n$ that are ...
0answers
64 views

### Does $\ 3\mid (M_p - 4)$?

So I was given a challenge by my maths teacher and I do not know how to solve it: Challenge: Find three prime numbers $p, q, r$ such that $pqr(p + q + r) = P = \text{Perfect Number}$. ...
0answers
38 views

### How can I prove that $1 + 2 + \cdots + x = \{P : P = \text{Perfect Number}\} \Leftrightarrow 2^n - 1 = x = \{M_p : n\in \mathbb{N}\}?$

I looked at formulas concerning prime numbers, and I came across Mersenne Primes which are primes of the form $2^n - 1 : n\in \mathbb{N}$ and the symbol used to denote a mersenne prime is $M_p$. ...
0answers
71 views

### What are some basic properties of the quotient $s_n\over M_p$ where $M_p$ is a Mersenne prime?

I've been exploring the Lucas-Lehmer test for a while now. I already know the square of one Mersenne $(2^n-1)^2$ is $(2^{n-1}-1)(2^{n+1}-1)+2^{n-1}$. Today I'm looking to use the properties of the ...