Unsolved Problems

Show that the equation $ n! = a_1!a_2!\cdots a_k!, $ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions....

For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots <a_k=m$ with $a_1!\cdots a_k!$ a square....

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

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 restric...

We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count ...

Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can...

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \prod_{0\leq i\leq k}(p^2+i) $ is $p$?...

Let $ F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m), $ where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all s...

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. $...

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 $(cn,n]$?...

Can one classify all solutions of $ \prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there...

Is it true that for every $n\geq 1$ there is a $k$ such that $ n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)? $ ...

Let $f(n)$ be the minimal $m$ such that $ n! = a_1\cdots a_k $ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $ f(n)-...

Let $f(n)$ denote the minimal $m\geq 1$ such that $ n! = a_1\cdots a_t $ with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?...

Let $t_k(n)$ denote the least $m$ such that $ n\mid m(m+1)(m+2)\cdots (m+k-1). $ Is it true that $ \sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c} $ fo...

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

For any $k\geq 2$ let $g_k(n)$ denote the maximum value of $ (a_1+\cdots+a_k)-n $ where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid ...

For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots<a_n$ with $ p^k \mid (a_1!+\c...

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?...

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1...

How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of...

Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for all $n\geq 2$ $ \lim_{k\to \...

Let $g_1=g(n)=n+\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?...

Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that, for every $m,n\geq 2$, there ex...

Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\omega(m) \leq n$...

Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist ...

For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)...

Let $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $V(2x)/V(x)\to 2$? Is there an asymptotic formula for $V(x)$?...

Let $ V'(x)=\#\{\phi(m) : 1\leq m\leq x\} $ and $ V(x)=\#\{\phi(m) \leq x : 1\leq m\}. $ Does $\lim V(x)/V'(x)$ exist? Is it $>1$?...

If $\tau(n)$ counts the number of divisors of $n$ then let $ F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}. $ Is it true that $ \lim_{n\to \i...

Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct?...

Let $f(1)=f(2)=1$ and for $n>2$ $ f(n) = f(n-f(n-1))+f(n-f(n-2)). $ Does $f(n)$ miss infinitely many integers? What is its behaviour?...

Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the se...

Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i eq j$. Is it true that th...

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a cons...

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and $ \liminf\frac{\lv...

Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1}...

Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?...

Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?...

If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let $...

Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pm...

How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?...

Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$....

Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\...

Let $ f(n) = \min_{i<n} (p_{n+i}+p_{n-i}), $ where $p_k$ is the $k$th prime. Is it true that $ \limsup_n (f(n)-2p_n)=\infty? $ ...

Let $q_1<q_2<\cdots$ be a sequence of primes such that $ q_{n+1}-q_n\geq q_n-q_{n-1}. $ Must $ \lim_n \frac{q_n}{n^2}=\infty? $ ...

Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $m_n<p_n$ for al...

Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide $ \prod_{1\leq i\leq \log n}(n+i...

Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq i<k$. Does $ \sum_{...

Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)...