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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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Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$

Somewhere on Stack Exchange I saw the equation $$\gcd(2^m-1,2^n-1)=2^{\gcd(m,n)}-1.$$ I had never seen this before, so I started trying to prove it. Without success... Can anyone explain me (so actually prove) why this equation is true? And can we…
Bart Michels
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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 $p > 2$,…
DaBler
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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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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 $6n+1$ ,…
Pedja
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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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How many numbers in the interior of Pascal's triangle are Mersenne numbers?

Consider the interior of Pascal's triangle, i.e. the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. How many numbers in the interior of Pascal's triangle are Mersenne numbers, that is, numbers of the…
Dan
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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…
RParadox
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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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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 prime cannot divide an odd perfect number (quote:…
Jeppe Stig Nielsen
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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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Summation involving exponents of Mersenne primes

We all know the Leibniz formula for $\pi$ : $\dfrac{\pi}{4}=\displaystyle\sum_{k=0}^{\infty}\dfrac{(-1)^k}{2k+1}$ . Now observe the following summation: $$s=1+\displaystyle\sum_{p \in \mathbb{M}}\dfrac{(-1)^{(p-1)/2}}{p}$$ where $\mathbb{M}=\{p |…
Pedja
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Justification for the Sequence in the Lucas-Lehmer Primality Test for Mersenne Primes.

A Mersenne prime is a prime number of the form $M_p = 2^p -1$, for $p$ a prime number. We have the sequence $$s_i = \begin{cases} 4, \text{ if } i = 0;\\ s_{i-1}^2 - 2 \text{ otherwise. }\end{cases}$$ The Lucas-Lehmer primality test says that $M_p$…
YesYer
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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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