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I've been thinking about such an interesting question these last few days. I wonder if anyone has studied it. Question: Find all postive integers $n$ such that $$S(n)=1^n+2^n+3^n+4^n+\cdots+n^n …

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 $\dfrac{2^n-2}{n}$ …

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 …

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 …

Following problem though not a research problem if $x,y,z,w$ are postive integers,and such $$xyzw=504(x^2+y^2+z^2+w^2)$$ such example $(x,y,z,w)=(21,63,84,84)$ hold, Now My problem there exist …

Conjecture Let $m$ be give positive integer numbers,a sequence $p_{1},p_{2},\cdots $of primes satisfies the following condition:for $n\ge 3$,$p_{n}$ is the greatest prime divisor $p_{n-1}+p_{n-2}+ …

Cross-Posted from Math Stackexchange Two positive integers $p,q$ and a prime $r$ are given, such that $r>p>q>1$. I have to show that there exist $n$ such that $$r|\binom{p^n}{q^n}$$ Should I use Luc …

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. …

Question: Assume that $a,b\in Z$, and $4a^3+27b^2\neq 0$. Prove that there exist infinitely many positive integers $n$ such $n^3+an+b$ is square-free. I have following There exist infinitely m …

Prove that for all positive integers $ n$ different from $ 3$ and $ 5$, $ n!$ is divisible by the number of its positive divisors. I tried some things,such as the number of divisors of $ n! …

@Allan methods it's nice! here is my answer: since $$\left(\dfrac{1-a^2}{2a}\right)\left(\dfrac{1-b^2}{2b}\right)\left(\dfrac{1-c^2}{2c}\right)=1$$ so let $$\dfrac{x}{y}=\dfrac{1-a^2}{2a},\;\dfrac{y}{ …

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: Let $m$ be a positive integer. Then $$f(m)=(2m)^{2m+1}+m^{2m+1}\cdot (2m+1)^m+(2m+1)^{2m}$$ is not a prime number. One can prove it when $m$ is odd number, it is clear that $f(m) …

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

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 …