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Results tagged with prime-numbers
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user 38851
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
0
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0
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240
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On $n$-th prime $\pmod {n}$
Has it been proved or disproved that for any fixed $a\geq 1$ there are infinitelly many primes $p_n\equiv a\pmod{n}$?
I believe i have proved that for every $a\geq1$ there are infinitelly many natur …
- 3,866
3
votes
1
answer
240
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Is $2^n -1$ finitely many times the product of consecutive primes? [duplicate]
This question was asked at MSE but recieved no attention at all.
Here it is:
Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ?
$p_1=3,p_2=5 , ...,p_k$ are consecu …
- 3,866
1
vote
1
answer
308
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Radical of the sum $=$ radical of the product
My question is:
Has it been proved/disproved or studied the following?
For every $k\geq 4$ there are $k$ pairwise relatively prime numbers $a_1,a_2,...,a_k$ all greater than $1$ such that $$\t …
- 3,866
11
votes
3
answers
747
views
Does $2^n-n$ have infinitely often a prime divisor greater than $n$?
I think the question in the title is clear.
Let $n\in \mathbb{N}$. It is a nice exercise to show that every prime number divides infinitely many terms of the sequence $2^n-n$. (For example take $n=(p- …
- 3,866
10
votes
1
answer
700
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Primes dividing $2^a+2^b-1$
From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$.
Is it possible to prove that there are infinitely many primes not
dividing $2^a+2^b-1$?
(With $2^a …
- 3,866
3
votes
When is $a^{2^n}+1$ prime finitely often unconditionally?
I am almost copy - pasting a book of Sierpinski right now (I think the latest version of it)
"A. Schinzel showed that for every $a\in \mathbb{N}$ with $1<a<2^{27}$
there is an $n\in \mathbb{N} …
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2
votes
Why such an interest in studying prime gaps?
A new (strong) result may affect proving Legendre's conjecture
The first thing you can read there is "prime gaps".
- 3,866
10
votes
0
answers
221
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Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \qua …
- 3,866
8
votes
1
answer
415
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Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime
Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?
It seems that for every odd $m$ there are many $(a,b)\in \mathb …
- 3,866
6
votes
3
answers
822
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Is it possible to multiply two series to get as a result all composite numbers?
I was toying with the following problem:
Is it possible to find two infinite integer sequences $(a_n), (b_n)>0$ such that $\sum_{n=1}^{\infty}\frac{1}{(a_n)^s}\cdot \sum_{n=1}^{\infty}\frac{1}{(b_n)^s …
- 3,866
7
votes
3
answers
1k
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a question for the prime counting function
A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that
$\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$.
Using this inequality we can prove that wh …
- 3,866
2
votes
2
answers
800
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prime zeta function when $0<s<1$ [closed]
I will not be surprised if this question seems trivial in MO but I asked it first in MathSE and I did not get an answer. Here is the question:
I would like to know if there is a good estimate for th …
- 3,866
16
votes
1
answer
1k
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Divergence of a series similar to $\sum\frac{1}{p}$
Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic …
- 3,866
1
vote
0
answers
71
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On the sum of digits of primes in binary form [duplicate]
Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has $s_ …
- 3,866
1
vote
0
answers
98
views
Reference request for a result in additive combinatorics
Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally distribut …
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