# Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

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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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### 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 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. $2^{11}... 1answer 80 views ### The sum of all divisors of$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 ... 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 ... 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 ... 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 ... 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, ... 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 ... 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$2$. ... 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$. The ... 2answers 118 views ### Disprove: If$\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 ... 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 ... 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)(...
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 ...