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For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that $$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$ Let \begin{al …

Question: let $k$ be a positive integer, $p$ a prime number, such that $p=3k+1$, $r<p$ be a positive integer, such that $2^{k+1}\equiv r\pmod p$, and $r\not\equiv 4,5\pmod 6$. Show that $$p\nmid k!+1. …

Let $f(x)$ be an integer-valued polynomial (when $x\in \mathbb{Z}$, then $f(x)\in \mathbb{Z}$), and $a,b$ be positive integers, and $p$ be a prime number with $(a,p)=1$. Show that $$\sum_{x=0}^{p-1}\l …

$\newcommand\Legendre{\genfrac(){}{}}$Let $p\equiv 1\pmod 4$ be a prime number, and $x_{i}\ge 0$ be such that $$x_{1}+x_{2}+\dotsb+x_{p}=1.$$ Show that $$\sum_{1\le i<j\le p}\Legendre{i-j}{p}x_{i}x_{ …

Are there infinitely many postive integers $\ n\ $ satisfying $$\varphi(n)\mid \sigma(n)$$ where $\varphi(n)$ is Euler’s totient function, and $\sigma(n)$ is the sum of divisors of $n$? If yes, can …

If positive integer $n$ such $n\mid2^n-2$,where $n>1$, we called $n$ is Poulet number, see: https://en.wikipedia.org/wiki/Super-Poulet_number I found if $n>2$ is Poulet number, then $\dfrac{2^n-2}{n}$ …

Find the positive integers $(2^x-1)(3^y-1)=2z^2$ have three solutions $$(1,1,1),(1,2,2),(1,5,11)$$I already know $(2^x-1)(3^y-1)=z^2$ has no solution. See: P.G.Walsh December 2006 [On Diophantine equ …

Conjecture：Today I have no intention of thinking about this question. I have only got two solutions so far. I guess there are only two solutions, but I won't prove it. Let $n$ be positive integers, s …

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ we have Prove that there exists …

One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$. But does there exist a simple w …

Let $a,b,c$ be poistive integers,and such $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$,fine the all $a,b,c$ such $$a^2+3b^2c^2=7^c$$ I'm not sure that this question has been studied, but I've been trying for a l …

if $n>k>1$ be postive integer,show that $$S_{k}(n)=\dfrac{1}{n^k}\sum_{j_{1}=1}^{n}\sum_{j_{2}=1}^{n}\cdots\sum_{j_{k}=1}^{n}\gcd(j_{1},j_{2},\cdots,j_{k})\le\dfrac{\zeta(k-1)}{\zeta(k)} \tag{1}$$ …

Let $m,n$ be positive integers, such that $m^2\neq n^3$. I conjecture the inequality $$|m^2-n^3|>\dfrac{1}{5}\sqrt[6]{m^2+n^3}.$$ I've tried a lot of numbers, and they all seem to work, but how do I …

The problem comes from a problem I encountered when I wrote the article Find all positive integer $m$ such $$2^{m}+1\mid5^m-1$$ it seem there no solution. I think it might be necessary to use qu …

By Fermat Last theorem, I don't know if that's been discussed. Find all positive integers $x,y,z$ such $$x^3+y^3=3z^3$$