Školjka

%V0 Define the sequence $a_1, a_2, a_3, ...$ as follows. $a_1$ and $a_2$ are coprime positive integers and $a_{n + 2} = a_{n + 1}a_n + 1$. Show that for every $m > 1$ there is an $n > m$ such that $a_m^m$ divides $a_n^n$. Is it true that $a_1$ must divide $a_n^n$ for some $n > 1$?

%V0 Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

%V0 Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

%V0 The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: $a_0=2, \quad a_{k+1}=2a_k^2-1 \quad$ for $k \geq 0$. Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. comment Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u Edited by Orl.

%V0 Let $k$ be a positive integer. Show that if there exists a sequence $a_0$, $a_1$, ... of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}$$ then $k-2$ is divisible by $3$. Proposed by Turkey

%V0 Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $\left(a^n-1\right)\left(b^n-1\right)$ is not a perfect square. Proposed by Mongolia