Unsolved Problems

Classify graphs with representation number exactly 3....

Among bipartite graphs, do crown graphs require the longest word-representants?...

If every edge has imbalance ≥1, is the multiset of edge imbalances always graphic?...

Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?...

Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?...

Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?...

Do Lychrel numbers exist in base 10?...

Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?...

Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it ...

Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - ...

Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \le i \le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0...

Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?...

Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \dots + (n-1)^{n-1} + 1$, then $n$ is prime?...

In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of...

Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \le 2^k < n$?...

Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the dia...

Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?...

Does there exist a composite number $n \equiv 3$ or $7 \pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?...

Does there exist an even Fibonacci pseudoprime?...

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $ N \gg ...

Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?...

Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?...

(Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-t...

(Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a met...

Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble....

(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can...

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable....

(Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators?...

(Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2?...

(Well-known problem). Does there exist a finitely presented group $G$ such that $G\cong G\times H$ for some non-trivial group $H$?...

Let $\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble gro...

A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\leqslant G_1\leqslant\cdots\leqslant G_n=G$ such that for every...

Let $\Gamma$ be a finite simple group and let $N_n(\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is i...

Conjecture: For $n\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$....

Conjecture: Metabelian groups are permutation-stable....

A group $G$ is said to be sofic if for every finite set $F\subseteq G$ containing $1$ and every $\varepsilon>0$ there exist $n\in\mathbb N$ and a map ...

Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$....

Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $...

Conjecture: If $G$ is a transitive permutation group on a finite set $\Omega$, then for any distinct $\alpha,\beta\in\Omega$ there is an element $g\in...

For a group word $w(x_1,\ldots,x_n)$ on $n$ letters, define $e_0(x_1,\ldots,x_n)=x_1$ and $e_{k+1}(x_1,\ldots,x_n)=w(e_k(x_1,\ldots,x_n),\ldots,x_n)$ ...

Conjecture: The derived length of a finite solvable group $G$ does not exceed $|\operatorname{Cod}(G)|-1$....

Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\alpha(x)$ be the smallest number of conjugates of $x$ in...

Conjecture: Let $G$ be a finite additive abelian group with $|G|$ odd. Then any subset $A$ of $G$ with $|A|=n>2$ can be written as $\{a_1,\ldots,a_n\}...

(Well-known problem). Is Thompson's group F automatic?...

Conjecture: Every subgroup of Thompson's group F is either elementary amenable or else contains a subgroup isomorphic to F....

(Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial....

Is the crossing number additive under connected sum: $c(K_{1}\#K_{2}) = c(K_{1}) + c(K_{2})$?...

(a) Show that if $P$ is a nontrivial satellite operator and $K_{P}$ is a nontrivial satellite of a knot $K$, then $$ c(K_{P}) \geq c(K), $$ where $c...

How does unknotting number behave under connected sum and mutation? (a) Does the connected sum of $n$ nontrivial knots have unknotting number at least...

Let $P$ be a nontrivial satellite pattern with winding number $w(P) \neq 0$. Then for any nontrivial knot $K$ and its satellite $K_{P}$ , one has $$ ...