All Questions
11
questions
3
votes
1
answer
217
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 ...
- 141
3
votes
1
answer
364
views
prime number (a form like Mersenne primes)
I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a ...
2
votes
0
answers
105
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. ...
- 1,024
2
votes
3
answers
185
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:
...
- 163
1
vote
2
answers
108
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 ...
- 2,566
1
vote
1
answer
146
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 -...
- 13
2
votes
0
answers
105
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 ...
- 9,299
1
vote
0
answers
81
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 ...
- 6,014
2
votes
1
answer
184
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 ...
user243301
1
vote
1
answer
504
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, $...
- 35
5
votes
1
answer
173
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 ...
- 908