Questions tagged [mersenne-numbers]

For specific number theory question related to Mersenne numbers.

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Proving that for every prime number $p$, the number $2^p - 1$ is either prime or divisible by a prime $q$ such that $q \equiv 3 \pmod{4}$. [duplicate]

Is it true for every prime number $p$, the number $2^p - 1$ is either prime or divisible by a prime $q$ such that $q \equiv 3 \pmod{4}$?
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1 vote
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New Mersenne Primes Found -- Need help with Lucas-Lehmer Tests Please. [closed]

I wrote some Python3 code last night to work on this problem for fun and seemingly stumbled upon two new Mersenne Primes. I'm still learning so I don't quite understand how to format the Lucas-Lehmer ...
• 19
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Let p and q be prime numbers with $p \equiv 3 \pmod 4$ and $q=2p+1$. Prove that $2^{p} - 1$ is prime only when $p=3$

Task: Let p and q be prime numbers with $p \equiv 3 \ \pmod 4$ and $q=2p+1$. Prove that $2^{p} - 1$ is prime only when $p=3$. This question has been asked on this forum for a long time, but I have ...
153 views

Can it be proven that the reduced digit sum of Mersenne primes, except 3 and 7, is always 1 or 4?

For the first 36 Mersenne numbers (M(n)) I have calculated M(n)(mod 9) and the reduced digit sum of M(n). For n > 0, I found repeating sequences of respectively 1,3,7,6,4,0 and 1,3,7,6,4,9. Also, I ...
• 21
1 vote
99 views

About divisors of Mersenne numbers [closed]

1. Any Mersenne number $M_p,~p ≡ 3 \pmod 4$ where $p$ is a prime, can be represented as: $$(8px+2p+1)(8py+1) = M_p,~0 ≤ x ≤ \frac{M_p - 1 - 2p}{8p},~ 0 ≤ y ≤ \frac{\frac{M_p}{2p + 1} - 1}{8p}$$ 2. Any ...
1 vote
138 views

$2^n + 3$ and $2^{n+1} + 3$ both prime?

It seems there are infinitely many primes of the form $2^n + 3$. It seems that the density is about the same density as primes below $n$ at least for small $n$. What surprised me is I found solutions ...
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342 views

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 ...
• 23.9k
206 views

A question about prime factorization of composite Mersenne numbers and $(2^p-2)/(2 \cdot p)$

Mersenne numbers are numbers of the form $2^p-1$ where $p$ is a prime number. Some of them are prime for exemple $2^5-1$ or $2^7-1$ and some of them are composite like $2^{11}-1$ or $2^{23}-1$. I'm ...
• 131
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Is there a positive integer $n$ such that every $n$-Mersenne prime generates a prime?

First, some definitions. A $1$-Mersenne prime is just a Mersenne prime. Now, let $n$ be a positive integer. If $p$ is an $n$-Mersenne prime, and $2^p - 1$ is itself a prime, then $2^p - 1$ is said to ...
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A Mersenne number is never a Carmichael number

I am tasked with proving that all Mersenne composites (that is, composite numbers of the form $2^n -1$) are either always Carmichael numbers or never are. Running some tests, I have found some ...
• 345
1 vote
218 views

An explanation of why the Lucas-Lehmer Primality Test works: is this OK?

I've always wondered why the Lucas-Lehmer Primality Test works. After studying it, I came up with an explanation. I hope you can help me confirm and complete it. A Mersenne prime is a number of the ...
135 views

Decimation of Maximal-length sequence

Consider a Maximal-length sequence (hereby M-sequence) $S$ of minimal period $2^n-1$, as produced by an LFSR with a binary primitive polynomial. Note $s_0,s_1,\ldots$ the bits of $S$. For some integer ...
• 1,768
78 views

Mersenne number variant

It appears that all numbers of the form $(6^p-1)/5$ for prime $p$ are square free and are coprime to each other. Is this true? This seems like it might be related to Mersenne numbers, except with 6 ...
• 21
60 views

Generalization of Mersenne Primes

Besides Mersenne primes, are there any other known conditions for the natural numbers $n$ and $m$ under which the natural number ${n^m - n + 1}$ is "unusually likely" to be prime? By ...
• 191
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Squarefreeness Mersenne Numbers

In this page of Caldwell, after the proof of the link with Wieferich primes and pointing out that the two known ones can't be divisors of any Mersenne numbers, in the Comment he asserts «... so $M_q$ ...
• 111
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Numbers of divisors for a Mersenne number

Recently I encountered a problem: If n is a positive integer, then is the number of divisors of $2^n - 1$ less or greater than the number of divisors of n? I tried factoring and taking modulo n but ...
• 21
1 vote
77 views

Use Mersenne numbers to prove that there are infinitely many prime numbers.

When reading the book Mersenne Numbers and Fermat Numbers, after proving that: for any positive integers m,n, it holds $\gcd(M_n,M_m)=1$ if and only if $\gcd(m,n)=1$, it says that this allows us to ...
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• 363
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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 ...
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Subarray Sum Equals K - If the same prefix sum is encountered why does it follow 1,3,7,15,31 pattern?

This is an interesting property I have noticed in the problem: Subarray Sum Equals K The basic algorithm is as follows: You start with calculating the prefix (running sum). You check if the ...
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Odd Composite Proof

When $n$ is an odd natural composite number, $\frac{2^n + 1}{3}$ is always composite. Prove this conjecture. I'm not 100% sure but I let $n$ = $2c+1$ = $ab$, where $c$ is a natural number including $0$...
1 vote
244 views

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A note on conjecture that all the Mersenne numbers are square-free

Today, I am posting an approach that seems too elementary to validate. The approach is developed by me while reading the 1967 work of L. Warren and H. Bray. In their article, "On the square-...
• 191
1 vote
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A diophantine equation inspired in a conjecture due to Gica and Luca

In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$ over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and Mersenne exponents $x\geq 13$ such that $x^2-2$ is ...
1 vote
98 views

Assume that there are infinitely many Mersenne primes, prove there are infinitely many positive integers m and n such that $\phi(m)=\sigma(n)$.

Conjecture 3 of chapter 1 states that there are infinitely many Mersenne prime numbers. Mersenne prime numbers are integers that are in the form of $2^p-1$ where p is prime.The first 3 Mersenne prime ...
1 vote
101 views

Is this a valid proof concerning Mersenne Primes?

I am very new to proofs, and I learned that for $2^c-1=n$, if $c$ is composite, then $n$ is composite. I recently learned about proofs by contradiction, so here is my attempt at proving this ...
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