Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

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4
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1answer
63 views

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 ...
0
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0answers
30 views

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 ...
3
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2answers
107 views

Prove that number of times $3$ divides $2^n\pm1$ is exactly one more than the number of times $3$ divides $n$

TL;DR How to prove the eight congruences at the end of this post? Remark. My number theory is rusty and I'm trying to prove the following observations. Motivation: This result easily implies that $3^...
-1
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1answer
44 views

Let $x=2^{p-1}-1$ be composite, $p$ prime and $3\mid x$. Why $p \mid x$?

Let $x=2^{p-1}-1$ be a composite odd natural number (a "wrong" Mersenne) and $p$ is prime and $3\mid x$. Why $p \mid x$? Does really $p$ always divides $x$? Note: We know that $x$ is a sum of powers ...
1
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0answers
70 views

About the probability that the number of Fermat primes is infinite. [duplicate]

I am asking about the probability that the number of Fermat primes is infinite. There is a lot of things similar to the case of Mersenne primes. But it was conjectured that the number of Mersenne ...
-1
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1answer
67 views

There exists infinitely many primes formed of 2^(F[n!+1])-1

I have a conjecture: Ultra-Primes-Conjecture. There exists infinitely many primes formed of $2^{F[n!+1]}-1$. Here $2^p-1$ is Mersenne number, $F[n+2]=F[n+1]+F[n], 0,1,1,2,3,5,8,...$ is Fibonacci ...
2
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1answer
77 views

Property of some composite Mersenne numbers

I noticed this property of some composite Mersenne numbers: If $p$ is prime, $p=1 \bmod 4$ and $(1+6 \cdot p)$ is prime for a theorem of Fermat every prime $p=1 \bmod 3$ can be written as $p=c^2+3 \...
-2
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1answer
145 views

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.
1
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1answer
79 views

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.
2
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0answers
90 views

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. ...
-1
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1answer
201 views

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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0answers
22 views

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 ...
2
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3answers
96 views

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 ...
0
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2answers
50 views

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....
1
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1answer
57 views

For relatively prime $a$ and $b$, if a prime $p$ divides $a^n-b^n$ then we one of two cases

For relatively prime positive integers $a$ and $b$ and a natural number $n$, if a prime $p$ divides $a^n-b^n$ then either $p$ divides $a^d-b^d$ for some divisor $d$ of $n$ such that $d<n$, or $p\...
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4answers
78 views

Mersenne primes Lemma [duplicate]

How does one show that if $2^n-1$ is a prime, then $n$ is a prime?
1
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2answers
84 views

Mersenne primes and the sequence $w_0=2,w_{n+1}=2w_n^2-1$

Given the sequence $w_0=2, w_{n+1}=2w_n^2-1.$ It appears, from purely numeric examples, that we have this result: When $q$ is an odd prime number, then $M_q=2^q-1$ is prime if and only if $M_q\...
2
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3answers
116 views

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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0answers
58 views

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 ...
0
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1answer
68 views

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 $K$,...
1
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1answer
79 views

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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2answers
180 views

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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1answer
140 views

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 ...
2
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3answers
145 views

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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2answers
75 views

$\frac{q^p-1}{q-1}$ 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 ...
10
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1answer
242 views

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
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1answer
128 views

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&...
1
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1answer
96 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 ...
1
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1answer
91 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 -...
2
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0answers
114 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, ...
2
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1answer
57 views

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}...
1
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1answer
80 views

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 ...
2
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0answers
54 views

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 ...
1
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2answers
76 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 ...
2
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0answers
77 views

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 ...
0
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2answers
94 views

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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0answers
65 views

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 ...
1
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1answer
127 views

$b^{n}+1$ 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 $2$. ...
3
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2answers
60 views

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 ...
4
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2answers
118 views

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 ...
2
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1answer
148 views

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 ...
3
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2answers
78 views

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)(...
4
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1answer
172 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 ...
4
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1answer
58 views

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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0answers
130 views

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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0answers
155 views

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 ...
2
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0answers
151 views

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 ...
2
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0answers
96 views

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 ...
2
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0answers
57 views

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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0answers
27 views

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 ...