Unsolved Problems

Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq m...

Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that {UL} {LI} $A\backslash B$ is complete for any finite subset $B$ and {/LI} {LI} $A\bac...

For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that {UL} {LI} $A$ remains complete after ...

For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum...

Let $p(x)\in \mathbb{Q}[x]$. Is it true that $ A=\{ p(n)+1/n : n\in \mathbb{N}\} $ is strongly complete, in the sense that, for any finite set $B$, $ ...

Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?...

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset $ \{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloo...

Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $...

Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $ n=\sum_{u\leq i\leq v}a_i. $ Is there such a...

Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlie...

Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a s...

Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square...

Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that,...

How large is the largest prime factor of $n(n+1)$?...

Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ a...

Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$....

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...