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Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

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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 ...
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Number of Collatz steps for Mersenne numbers

I noticed that for all $k \in \mathbb{N} \geq 1$ the following is true (I tested up to $2^{5000}$): $\text{Collatz_Steps}(2^{2k+1} - 1) + 1 = \text{Collatz_Steps}(2^{2k+2} - 1)$ Where $\text{...
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$\frac{q^p-1}{q-1}$ squarefree?

Is $\frac{q^p-1}{q-1}$ always squarefree with $q,p$ prime and $p>2$ and $(q,p)=(3,5)$ excluded? This is a follow up of $3^p-2^p$ squarefree? I know the case $q=2$ (Mersenne) and $q=3$ are still ...
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New property of 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 ...
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Nested Mersenne Primes

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&...
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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 ...
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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 -...
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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, ...
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Do we know a $2^{M_i}-1$ that's not prime?

Do we know a $2^{M_i}-1$ that's composite, i.e. not prime, where $M_i$ is a Mersenne prime number of the form $2^p-1$? For example $2^7-1=127$ is not an example because 127 is actually prime. $2^{11}...
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The sum of all divisors of $N$ is a 2 potency iff $N$ is a product of different Mersenne primes

As far as I have controlled: $\sigma(a)=2^n$, for some $n\in\mathbb N \iff $ $a$ is a product of different Mersenne primes. The $\Leftarrow$-part is an immediate consequence of that $\sigma$ is ...
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Is any ec-number larger than $73$ the sum of a square and a cube?

An ec-number concatenates the Mersenne numbers $2^n-1$ and $2^{n-1}-1$ decimal (see also here A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal ...
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53 views

Is it possible the density of Mersenne numbers in the primes gets arbitrarily close to $1$?

Let a Mersenne number be $M_p=2^{p}-1:p\in\text{prime}$ Suppose above some lower bound every Mersenne number were prime and every prime number were a Mersenne number. Can this conjecture be shown ...
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What are the most (and least) likely factors of a composite Mersenne number?

What are the most (and least) likely factors of a composite Mersenne number? Suppose some number $M_p=2^p-1:p\in\text{prime}$ is a candidate for a Lucas Lehmer test. Is it possible to identify a set ...
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Are Mersenne numbers with Mersenne prime exponent always prime?

A Mersenne number is a number on the form $2^n-1$. For it to be prime, the number $n$ must be prime. My question is that if $n$ is another Mersenne prime, will $2^n-1$ be always prime? It seems so, ...
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Generating prime factors of a certain congruence?

I'm aware that prime factors of $n^2+1$ take the form $4k+1$. It's also well known that factors dividing $\frac{a^p \pm 1}{a \pm 1}$ will be congruent to $2kp+1$. Fibonacci and some other recurrence ...
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$b^{n}+1$ is prime only if $b\equiv0\pmod2$ and $n$ is a power of $2$

Is there any known proof of the following conjectures: $b^{n}-1$, $n>1$, is prime only if $b=2$ and $n$ is prime. $b^{n}+1$, $n>1$, is prime only if $b\equiv0\pmod2$ and $n$ is a power of $2$. ...
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When $q$ is prime and a Wieferich prime $p$ divides $\ M=2^q-1\ $ . Why can we conclude $\ p^2\mid M\ $?

Here : https://en.wikipedia.org/wiki/Wieferich_prime I came across the following claim : A prime divisor $p$ of $M_q$, where $q$ is prime, is a Wieferich prime if and only if $p^2\mid M_q$. The ...
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Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree.

Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree. Equivalently, if $2^n-1$ is not squarefree, then $\gcd(n,2^n-1)\neq 1.$ Exercise, which I do,says to show that statement above is false. I ...
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The equation $\varphi(n)=n-\log_2(1+\operatorname{gpf}(n))-\operatorname{gpf}(n)+1$ and Mersenne primes

Let $n\geq 1$ an integer, we denote the Euler's totient function as $\varphi(n)$ and the greatest prime dividing $n$ as $\operatorname{gpf}(n)$ (that it the arithmetic function defined in the ...
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Show that for every $n$ there is a prime $p$ such that

Show that for every $n$ either there is a prime $p$ such that $p$ divides $n^2-1$ but not $n-1$, or $n$ is of the form $2^k-1$. If there is a prime $p$ which divides $n^2-1$ then it divides $(n + 1)(...
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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 ...
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On a conjecture that $P_n^{\,2}+5^2+2^k=(P_n-1)^2+l^2$.

I was looking at perfect numbers and came across something that might serve a little interesting. Denote by $P_n$ the $n^\text{th}$ perfect number, then there appears to always exist $k\in\mathbb{W}...
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How does GIMPS work and what are these iterations?

I downloaded GIMPS today just out of curiosity and have been running it. On my machine it is checking $M_{52898149}=2^{52898149}-1$. From what I could find on wikipedia I suppose that GIMPS uses ...
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What's the fastest free software to test primality of a Mersenne prime?

Mersenne primes are primes of the form $2^n-1$ for positive integer $n$. Currently, the most widely known (and employed) software for testing large Mersenne prime candidates is Prime95, which is also ...
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Equations involving the Euler's totient function and Mersenne primes

In this post we denote the Euler's totient function as $\varphi(n)$, first we show a claim related to Mersenne primes, see for example this Wikipedia and secondly we are going to ask a related ...
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A race between ec-primes and mersenne-primes. Who will win in the long run?

Here A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$. the ec-primes are introduced. They emerge by concatenating the ...
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Which primes can never divide an “ec”-number?

The "ec"-numbers (named after Enzo Creti) are defined as $$(2^{n+1}-1)\cdot 10^m+2^n-1$$ where $m$ is the number of digits in the decimal expansion of $2^n-1$. Or shorter, we concatenate the Mersenne ...
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Do distinct pairs $(m,n) $ generate distinct generalized ec-numbers?

A generalized ec-number (named by Enzo Creti) is a number that emerges when we concatenate two arbitary Mersenne-numbers ($2^1-1=1$ is allowed as well). For example, $77$ , $637$ or $13$ are ...
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Primes formed by concatenating the mersenne numbers from $2^2-1$ to $2^n-1$

Concatenate the mersenne numbers $2^2-1$ to $2^n-1$ and define $f(n)$ to be the emerging number , for example $$f(6)=\color\red {3}\color\green {7}\color\red {15}\color\green {31}\color\red {63}$$ We ...
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Is there a $31$-dimensional manifold with 496 differential structures?

Milnor found a $7$-dimensional sphere with 28 differential structures. Is there a $31$-dimensional manifold with 496 differential structures?
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Why $n=4$ the only integer satisfying :$-1+2^{n²+n+41}$ to be prime from $n=0$ to $n=40$?

Few research to know more about primality of Mersenne primes where the power of $2$ is Euler formula prime which it is defined as : $-1+2^{n²+n+41}$, it is well known that $n²+n+41$ is a prime ...
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What will/should be the intention or purpose to compute and count prime numbers in the gap defined by two consecutive and large Mersenne primes?

I would like to ask this soft question about prime numbers. I believe that (if I'm right) that the more largest known prime number, with few exceptions, (has the form) is a Mersenne prime $$2^p-1,$$ ...
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359 views

Find the next twin Mersenne prime pair in a Wilson’s primeth recurrence sequence

Define $p_{x}$ to be the $x$-th prime number (for example, $p_{15}=47$). Then define the recurrence $ a_{0}=1$ and $a_{n}=p_{{a_{{n-1}}}} \;\;\text{for}\;\; n>0$. This is Wilson’s primeth ...
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Mersenne Prime - why are these two definitions equivalent?

According to Wikipedia: If $n$ is a composite number then so is $2^n − 1$. ($2^{ab} − 1$ is divisible by both $2^a − 1$ and $2^b − 1$.) This definition is therefore equivalent to a definition ...
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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 ...
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Why do Mersenne primes only have prime inputs?

A Mersenne prime is a prime of the form $2^n-1$. Only when $n$ is a prime itself is there a chance that $2^n-1$ is a Mersenne primes. The largest primes discovered are almost always Mersenne primes. ...
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There is at least one number with $k$ different prime factors in Mersenne sequence, right?

The question(s) of infinitude of Mersenne primes (and semiprimes), that is, is there an infinite number of them in Mersenne sequence are, to the best of my knowledge, still unsolved. We could ask are ...
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Mersenne semiprimes

Is it known does the set of Mersenne numbers contain an infinite number of semiprimes? By the same procedure as with primes if $n=abc$ with $a,b,c>1$ is composite number then $2^{abc}-1$ is ...
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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 ...
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Primetest for Mersenne numbers

The following example shows a primetest for Mersenne numbers. It's based on two theorems: 1) The largest prime devider of a number n is $\lfloor\sqrt{n} \rfloor$. 2) Divisor of $M_{17}$ for ...
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Proving that $q \mid 2^{q-1} - 1$ and $q$ is of the form $q = 1+2kp$

Let a prime $p\ge 3$ and let $q$, a prime such that $q \mid 2^p - 1$. prove that $q \mid 2^{q-1} - 1$ and show that it has the form $q = 1+2kp$, $k\in\mathbb N$. Note: I have been asked to show ...
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Mersenne Primes using Mersenne Primes as n

I have been exploring the fascinating world of prime numbers, particularly Mersenne Primes, and have noticed an interesting pattern. It seems to me that $2^n - 1$ is prime as long as $n$ itself is a ...
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120 views

How did they find the currently world's biggest known prime number?

Just came to know about the currently world's biggest prime number: $$2^{74207281} − 1$$ I know that this number has about $22.3$ million digits but how did the "Great Internet Mersenne Prime ...
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Why are mersenne primes not wieferich primes?

I am a complete beginner and I am studying the relation between mersenne and wieferich primes. Wikipedia says A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if $p^2$ ...
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63 views

lucas lehmer test.

I asked to use the Lucas_Lehmer test to show that $2^{11} -1$ is prime i was wondering if there are any by hand examples of someone using this test on a mersenne prime that anyone knows of. I tried ...
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71 views

Find the perfect numbers of the product of two primes, $2^p-1$ and $2^{p-1}$

A number $n\in N$ Show that if $p$ is a prime, such that $2^p - 1$ is also a prime, a Mersenne prime that is, then $n = 2^{p-1}(2^p-1)$ is a perfect number. So I know that $n$ must be divisible by, $...
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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 ...
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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 $pqr(p + q + r) = P = \text{Perfect Number}$. ...
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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 $M_p$. ...
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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 ...