Unsolved Problems

Conjecture If $a_1,a_2,\ldots,a_{2n-1}$ is a sequence of elements from a multiplicative group of order $n$, then there exist $1 \le j_1 < j_2 \ldots <...

Conjecture For every $0 \le t \le n-1$, the sequence in ${\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewes...

For every finite multiplicative group $G$, let $s(G)$ ( $s'(G)$ ) denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a ...

Say that a set $S \subseteq {\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums. Conjecture There exists a fixed cons...

Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$. Assume we start with some number $n$ and repeatedly take ...

Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1....

Problem Does for every integer $N$ exist a covering system with all moduli distinct and at least equal to~ $N$?...

Problem (1) Prove that there exist infinitely many positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$ Problem (2) Prove th...

For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elemen...

Conjecture Let $G$ be an abelian group of odd order and let $A,B \subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered...

Problem Find an explicit formula for Frobenius number $g(a_1, a_2, \dots, a_n)$ of co-prime positive integers $a_1, a_2, \dots, a_n$ for $n\geq 4$....

Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $1$. The number $2$ appears onc...

Problem Find six positive integers $x_1, x_2, \dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do n...

Problem (1) Find a counterexample to Baillie-PSW primality test or prove that there is no one. Problem (2) Find a composite $n\equiv 3$ or $7\pmod{10...

Conjecture For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once. Equivalently: Conjecture For any prime $p>5$, $p^2$ doe...

Question Does there exist a $3\times 3$ magic square composed of distinct perfect squares?...

Conjecture Does a perfect cuboid exist?...

Conjecture Formulate a central limit theorem for the KPZ universality class....

Conjecture If $X_1, \dotsc, X_n \geq 0$ are independent random variables with $\mathbb{E}[X_i] \leq \mu$, then $$\mathrm{Pr} \left( \sum X_i - \mathbb...

Problem Is P = NP?...

Conjecture The famous 0-1 Knapsack problem is: Given $a_{1},a_{2},\dots,a_{n}$ and $b$ integers, determine whether or not there are $0-1$ values $x_{...

Problem Given two codes $R,C$, their Tensor Product $R \otimes C$ is the code that consists of the matrices whose rows are codewords of $R$ and whose ...

Problem Let $a_1,a_2,\ldots,a_n$ be natural numbers with $\sum_{i=1}^n a_i < 2^n - 1$. It follows from the pigeon-hole principle that there exist dist...

Question What is the complexity of the following problem? Given $a_1,\dots,a_n; k$, determine whether or not $\sum_i \sqrt{a_i} \leq k.$...

Problem Can $O(n)$-size circuits compute the function $f$ on $\{0,1,2\}^*$ defined inductively by $f(\lambda) = \lambda$, $f(0x) = 0f(x)$, $f(1x) = 1f...

If $p$ is prime and $g,h \in {\mathbb Z}_p^*$, we write $\log_g(h) = n$ if $n \in {\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an int...

Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, ...

Conjecture One-way functions exist....

Problem Prove unconditionally that $\mathcal{AM}$ $\subseteq$ $\Sigma_2$....

Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a de...

Conjecture For every rational $\epsilon > 0$ and every rational $\Delta$, there is no polynomial-time algorithm for the following problem. Given is a...

Question $S(S(f)) = S(f)$ for every endo-reloid $f$?...

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?...

Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?...

Problem Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates ...

Conjecture If a $4$-manifold has the homotopy type of the $4$-sphere $S^4$, is it diffeomorphic to $S^4$?...

Conjecture Given a knot in $S^3$ which is slice, is it a ribbon knot?...

Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^...

Problem Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-ba...

Problem Does there exist a subset of $\mathbb R^3$ such that its fundamental group has an element of finite order?...

Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smooth...

Problem $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeom...

Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalen...

Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?...

Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observa...

Question $( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f)...

Conjecture Let $C$ be a Cantor set embedded in $\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $...

Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property? A set has the fixed poi...

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group....

Conjecture $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \t...