# Questions tagged [prime-numbers]

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.

1,647 questions
Filter by
Sorted by
Tagged with
88 views

201 views

### Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$

Consider sum: \begin{equation} S_q(n) = \sum_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\} \end{equation} where $\{x\}$ is fractional part of $x$. It's easy to see that $S_{1}(n) = \frac{1}{2}(n-1)$, but ...
95 views

118 views

### Integers having the same digits as their prime factors

Say an integer is "digitally conservative" in base $b$ if the set of its digits coincides with the set of the digits of its prime factors, like $37127=137\times 271$ in base $10$, and denote ...
128 views

74 views

### Approximating real numbers [closed]

Given any two different prime numbers $p_1$ and $p_2$, any real number $r$ and an $\epsilon$. Can you always find two integers $n$ and $m$ such that $|\frac{p_1^n}{p_2^m} -r| \lt \epsilon)$? Here is a ...
241 views

### Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?

Note that for any permutation $\sigma\in S_5$ the product $\prod_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube. Question. Let $n>5$ be an integer. Is the product $\prod_{k=1}^nk^{\sigma(k)}$ ...
112 views

209 views

### The Diophantine equation $(xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers with $z > a^2+2a$

Problem. Let $a$ be a positive integer that is not a perfect cube. Show that the Diophantine equation $(xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers $x, y, z$ with $z > a^{2}+2a$. ...
53 views

### Primality of $n$ bit integers in depth $n^\alpha$ under standard conjectures?

Denote $\mathsf{NC}(\mathsf{SUBLINDEPTH}(n),f(n))$ to be set of boolean circuits of fan-in $2$ which can be represented by depth $\cap_{\alpha>0}\mathsf{}n^\alpha$ and $f(n)$ sized Boolean circuits....
214 views

### Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag*{(4)}$$ with $\operatorname{Ri}$ denoting the Riemann function ...
190 views

### Are there infinitely many solution $(a,b,p)$ satisfying $a^b+b^a=p$?

Let $a,b$ be integers which satisfy $2\leq a\leq b$ and $p$ a prime number. Are there infinitely many solution $(a,b,p)$ satisfying the following equation? \begin{eqnarray} a^b+b^a=p \end{eqnarray} I ...
123 views

133 views

### Small multiplicative order modulo infinitely many primes

Let $a>1$ be an integer. Artin's conjecture claims that if $a$ is not a perfect square, then $a$ is a primitive root modulo infinitely many primes $p$ (which moreover form a subset of positive ...
175 views

7k views

### What is the simplest proof that the density of primes goes to zero?

By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$. I want something much weaker, namely ...
The problems 'Given $0<a<b$ and a prime $p<a$ is there an integer $\ell\in[a,b]$ such that $p|\ell$?' 'Given $0<a<b$ and an integer $q\not\in[a,b]$ is there an integer $\ell\in[a,b]$ ...
The problem 'Given $0<a<b$ is there a prime in the interval $[a,b]$?' is in $\mathsf{NP}$. If we assume Cramer's conjecture the problem is in $\mathsf{P}$ since if $b-a>(\log a)^{2+\epsilon}$ ...