# Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

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### prime number (a form like Mersenne primes)

I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a ...
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### Is there a name for this family of sequences?

The sequence ${\displaystyle{M_n:=2^{p_n}-1}}$, where ${\displaystyle{n\gt0}}$ and ${p_n}$ is the ${\displaystyle{n}^{th}}$ prime number, is commonly known as the Mersenne numbers (not to be confused ...
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### On the equation $\psi(-1+2(\psi(n)-n))=n$ involving the Dedekind psi function, as a characterization of Mersenne primes

In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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### Show that $M_p^p\equiv 1 \mod p^2$

Can it be shown that $M_p^p\equiv 1 \mod p^2$ where $M_p=2^p-1$ is a Mersenne prime. I tried to develop the left part into into $2^{p^2}-1-pk2^p$ and use $2^{p^2}\equiv 2^p \mod p^2$, but I get ...
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### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes: reference request or proposal as an exercise

I don't know if the following question is in the literature, please add a commment if it is in the literature (I add my thoughts and motivation below in last paragraph, it is discursive and ...
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### About how much time would it take to test the primality of a billion digit Mersenne number with a typical processor?

I'm wondering how long it might take to run a Lucas Lehmer primality test on a one billion digit Mersenne prime using a 3.0 ghz processor.
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### Question about the exponents n in Mersenne Primes.

I am starting to study Mersenne primes, and I am wondering if there is a pattern in which exponents give rise to a Mersenne prime or if I am missing something. Thanks.
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### Can it be shown that numbers of a certain form produce primes more often than expected?

I am trying to figure out a way to measure if numbers of a given form are prime more often than expected. This would allow some way to quantify how useful certain forms are at producing large primes. ...
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### Different approach to solving the Collatz problem

Edit: A paper by Joseph Sinyor can be found here, and has a small section on The 3x+1 Problem and Mersenne Numbers, I think it is somehow relevant to what I was trying to deliver here. The ...
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### Getting factors of Mersenne numbers. Why does $2^p-1|2^b-1$ where $p|b$ and $p$ is prime and $b\in Z$? [duplicate]

Why does $2^p-1|2^b-1$ where $p|b$ and $p$ is prime and $b\in Z$? I'm working with Mersenne numbers and factorization of numbers to a large power minus 1 and the solution to the problems are to find ...
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### Mersenne prime variation

Mersenne primes are primes of the form $2^p-1$ where $p$ is some prime. I am wondering if primes of the from $q^p-2$ have been studied where $q>2$ is a prime and $p$ is also a prime. Are there ...
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### Let $p$ be prime number, and d is the natural number. Prove that if $d\mid 2^p−1$, then $p\mid d−1$

Let $p$ be prime number, and d is the natural number. Prove that if $d\mid 2^p−1$, then $p\mid d−1$. I'm looking on proof number 3 mentioned there and few things are unclear for me: https://en....
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### Lower Bound of a Factor of M = 2^P - 1, when M is a composite (P is prime).

I was wondering, is there any rule for the smallest factor of M (where M = 2^P - 1, P is a prime) when M is composite. I have an observation, I found the smallest factor for the following P: ...
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### Is There a Connection Between an Infinitude of Mersenne Primes and the Divergence of the Harmonic Series?

I recently came across https://primes.utm.edu/mersenne/ which asserts that there likely exists an infinite number of even perfect numbers because of the divergence of the harmonic series. Can ...
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### Are Semiprimes in the Mersenne Sequence Bound (Eventually) to Occur at Terms of Prime Index?

Related to a question posed a year and a half ago on the site, Mersenne semiprimes I would now like to ask, that in the sequence of Mersenne numbers, does there exist a bound on the indices, say $K$,...
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### The proof that the Mersenne number $M_{19}$ is prime.

Here is the hint to the proof given in the book: Using the following 2 theorems: 1-If $p$ is an odd prime, then any prime divisor of $M_{p}$ is of the form $2kp +1$. 2-If $p$ is an odd prime, then ...
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### Is $2^p - 1$ always prime when p is a mersenne prime?

First mersenne prime $2^2-1=3$, ** $2^{(2^2-1)}-1$ is also prime How many far can we go to get first composite? $2^{(2^{...(2^{(2^2-1)}-1)}-1)}-1$
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### Is there an Efficient Way to Divide by a Mersenne Prime?

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 ...
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A Mersenne Prime is any prime number of the form $2^n-1$, where $n$ is a positive integer. We can trivially see that for any Mersenne Prime $p=2^n-1$, $n$ has to be prime, as if $d \mid n$ and $1<d&... 1answer 145 views ### Fibonacci primes vs Mersenne primes It seems that only 34 Fibonacci primes are known while 54 Mersenne primes are known, while Fibonacci numbers are sparser than Mersenne numbers. Compare https://en.wikipedia.org/wiki/Fibonacci_prime ... 1answer 104 views ### Integers of this form that pass the Fermat Primality test are prime, proof? If an integer,$2p + 1$, where$p$is a prime number, is a divisor of the Mersenne number$2^p - 1$, then$2p + 1$is a prime number. My argument is that because divisors of the Mersenne number$2^p -...
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, ...