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Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$: $0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$. Prove that the sequence $(a_{n})$ is periodic. This …

Let $p(x)$ be a polynomial, $p(x) \in \mathbb{Q}[x]$, and $p^{(m+1)}(x)=p(p^{(m)}(x))$ for any positive integer $m$. If $p^{(2)}(x) \in \mathbb{Z}[x]$ it's not possible to say that $p(x) \in \mathbb{Z …

I asked this question at MSE, but I think it's more appropriated to MO. Let $x_{n}$ be a sequence, such that $x_{n+1}= \dfrac{nx_{n}^2+1}{n+1}$ and $x_n>0$ for all $n$. There is a positive inte …

Let $p(x) \in \mathbb{Z}[x]$, such that $\deg (p) \ge 3$. Can we always find $q(x) \in \mathbb{Z}[x]$, such that $\deg (q) < \deg(p)$ and $p(q(x))$ is reducible over $\mathbb{Q}[x]$? Is there a …

Let $a_1, a_2, \dotsc, a_k$ be $k$ positive integers, $k\ge2$. For all $n\ge n_0$ there is a positive integer $f(n)$ such that $n$ and $f(n)$ are relatively prime and $a_{1}^{f(n)}+\dotsb+a_{k}^{f(n)} …

Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n …

Are there infinitely many positive integers which are neither a sum nor a difference of two perfect powers? This question was proposed some years ago at KoMaL. It's easy to see that the odd nu …

Let $n \ge 2$ be a positive integer. Do there exist $n$ non-zero distinct integers such that the sum of their square is a perfect square and their product is a nth power? For $n=2$ the answer is no, b …

Let $P(x) \in \mathbb{Z}[x]$ be a monic polynomial of degree $n$, with no integer roots, such that $\upsilon_2(P(m))$ can assume $n$ different positive values, where $m \in \mathbb{Z}$ and $\upsilon_ …

The following question was asked at MSE without any solution: Show that the equation $x^3+y^3+z^3-2xyz=1$ have infinitely many integer solutions $(x,y,z)$. A more general question was also propo …

Let $k>1$ be a positive integer. Question 1. Does there exist infinitely many positive integers $n$ such that $$n \, | \, (1^n + 2^n + \cdots + k^n)?$$ And, more generally: Question 2. Does there e …

Is it true that for any positive integer $m$ there are infinitely many positive integers $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$? $\lfloor x \rfloor$ is the floor …

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$ and $k=p_1p_2 \ldots p_n$. It is a known result that $2^{k}+1$ has at least $4^n$ divisors (this was a shortlisted problem from IMO 2002 - …

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_{n}$ (i.e. $1 < d_{n} < P(n)$), such that the sequence $ …

Given a positive integer $m>1$, prove that there exists a polynomial of integer coefficients $P(x)$ that is irreducible on $\mathbb{Z}[x]$, has degree $2018$ and has a real root $r$ satisfying $m \mi …