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Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

Mersenne numbers are numbers of the form of $M_n = 2^n -1$. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those numbers whose binary representation contains a prime number of ones and no zeros. The smallest composite pernicious Mersenne number is $2^{11} − 1 = 2047 = 23 \times 89$.

Mersenne prime is a mersenne numbers which is a prime number. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes (sequence $A000668$ in the OEIS) are $3, 7, 31$, and $127$.

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For any fixed integer $ a \gt 1 $, how do you prove that $\frac{a^p-1}{a-1}$ is not always prime given prime $ p \not \mid a-1$?

I assumed this would be easy to prove but it turned out to be quite hard since the go to methods don't work on this problem. Once we fix any $a\gt 1$, we need an algorithm to produce a prime $p$ that makes $\frac{a^p-1}{a-1}$ composite and $a \not…
arbashn
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Is This a New Property I Have Found Pertaining to Mersenne Primes?

While playing with Mersenne numbers, I found the following property distinguishing Mersenne prime numbers from Mersenne composite numbers. A Mersenne number, $\text{M}p$, is a number of the form $2^p - 1$, where $p$ is prime. Property For $p > 2$,…
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How to show that all even perfect numbers are obtained via Mersenne primes?

A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). A well known theorem by Euler states that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime (this is what is called a Mersenne…
Gadi A
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How did Euler disprove Mersenne's conjecture?

In 1644, Mersenne made the following conjecture: The Mersenne numbers, $M_n=2^n−1$, are prime for $n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257$, and no others. Euler found that the Mersenne number $M_{61}$ is prime, refuting the conjecture. For…
actinidia
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Are there infinitely many Mersenne primes?

known facts : $1.$ There are infinitely many Mersenne numbers : $M_p=2^p-1$ $2.$ Every Mersenne number greater than $7$ is of the form : $6k\cdot p +1$ , where $k$ is an odd number $3.$ There are infinitely many prime numbers of the form $6n+1$ ,…
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Why are Mersenne primes easier to find?

9 out of 10 biggest known prime numbers are Mersenne numbers. Are they easier to find? rank prime digits who when reference 1 2**243112609-1 12978189 G10 2008 Mersenne 47?? 2 2**242643801-1 12837064 G12 2009 Mersenne…
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Mersenne primes before computers

On the Wikipedia page there is an ordered list of Mersenne primes and the dates they were discovered. The largest such primes and most recent discoveries were made with the help computers. But the largest Mersenne prime discovered without computer…
no lemon no melon
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Are there too many 8-digit primes $p$ for Mersenne primes $M_p$?

So it was recently announced that a new Mersenne Prime has been discovered: https://www.mersenne.org/primes/press/M77232917.html I was reading up a bit about Mersenne primes, and came across a conjecture of Lenstra–Pomerance–Wagstaff (LPW) on…
Kevin Ventullo
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Density of extended Mersenne numbers?

Consider the subset of odd positive integers defined and constructed as follows by these rules : A) $1$ is in the set. B) if $x$ is in the set , then $2x + 1$ is in the set. C) if $x$ and $y$ are in the set then $xy$ is in the set. I call them…
mick
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Divisibility of Mersenne numbers

Is there a way to prove that $2$ is the only prime that never divides $2^n-1$ ? Obviously we can ignore all primes that are themselves of this form. Some other examples: $$5\,|\,2^4-1 \qquad 9\,|\,2^6-1 \qquad 11\,|\,2^{10}-1 \qquad 13\,|\,2^{12}-1…
Christian
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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 prime. Now consider $$ F_n(k) = 2^n - k $$ with $k…
Fabian Schn.
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Prove that $2^{13}-1$ is prime

All prime divisors $p$ of $2^{13}-1=8191$ have $p\equiv 1\mod 26$. If $p$ divides $2^{13}-1$ then $2^{13}\equiv 1\mod p$, hence $2\in \Bbb F_p^\times$ has multiplicative order $13$. This gives us an order $13$ subgroup in $\Bbb F_p^\times$, hence…
MyNameIs
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Why do Mersenne numbers work?

In Matt Parker's book: "Things to make and do in the fourth dimension", he says that mathematicians have long known that for $2^n - 1$, if $n$ is not prime then the number cannot be prime. I don't understand how this works? $2^4-1=15$, and $4$ is…
Spammer
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Most efficient way to square modulo a Mersenne prime

I realise this question is somewhere in between Math StackExchange and StackOverflow. So forgive me if this is too much of a practical question, it would probably too theoretical elsewhere. I am looking for the quickest way to compute the square of…
Matteo Monti
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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 Dedekind psi function, and [1]. On the other hand I…
user759001
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