63 problems

Generalized Binomial Biroot Conjecture. The approximations converge according to

The q-log-convexity conjecture. (i) The sequence $(\sum_{k=0}^n\binom nk^3q^k)_{n\geq8}$ is $q$-log-convex; for every integer $m\geq2$, the sequence…

For a positive integer $r$, let $b_k(r)=\binom{k+r-1}{r}$, and let $R_n(b_k(r))$ denote the corresponding reciprocal Hankel matrix. Integrality criterion conjecture. The inverse of…

Let $m$ and $k$ be nonnegative integers, let $p$ and $q$ be parameters, and let $Z_p$ be the generator appearing in the $(p,q)$-deformed generalized Weyl algebra. Write $[i]_{p,q}$…

Are there only finitely many pairs of finite sets $M,N\subseteq\mathbb{N}$ such that $M$ and $N$ are disjoint and …

For $0\leq k\leq n$ write $\binom{n}{k} = uv$ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be the smallest $k$ su…

Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m mid \binom{2n}{n}$ satisfies $m\sim f(n).$

Let $S(n)$ be the largest integer $s$ such that, for every integer $k$ with $1\le k<n$, there is a prime $p$ (which may depend on $k$) for which $p^s$ divides the binomial coeffici…

Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?

Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$.

For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions.

For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficienc…

Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that $\binom{n}{k}=a$ (with $1\leq k\leq n/2$) has exactly $t$ solutions?

Are there infinitely many pairs of integers $n\neq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

Let $f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).$ - Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$.…

For every $n,i,j\in\mathbb N$ with $1\le i<j\le n/2$, does there exist a prime $p$ such that $i\le p$ and …

Does there exist a function $h:\mathbb N\to\mathbb N$ with $h(n)\to\infty$ as $n\to\infty$ such that, for all natural numbers $n,i,j$ satisfying $2\le i<j$ and $j\le n/2$, one has…

Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is…

Let $P(n,k)$ be the largest prime factor of $\binom nk$. Does there exist a real constant $c>0$ such that for every $n,k\in\mathbb N$ with $0<k<n$, …

Is it true that for every $k$ there exists $n$ such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$

Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.$

If $1<k<n-1$, must $\binom{n}{k}$ have a prime divisor smaller than $n/2$, with the sole exception $\binom{7}{3}=5·7$?

For fixed $r\geq0$, does the set of integers $n$ for which $\binom nk$ is squarefree for at least $r$ indices $1\leq k<n$ have a natural density, and is that density positive?

Is there some absolute constant $C>0$ such that $\sum_{p\leq n}1_{p mid \binom{2n}{n}}\frac{1}{p}\leq C$ for all $n$ (where the summation is restricted to primes $p\leq n$)?

Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?