Unsolved Problems

Conjecture There exists a finite lattice which is not the congruence lattice of a finite algebra....

Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic....

Let $R(k,k)$ denote the $k^{th}$ diagonal Ramsey number. Conjecture $\lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}}$ exists. Problem Determine th...

Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphi...

Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subva...

The deck of a graph $G$ is the multiset consisting of all unlabelled subgraphs obtained from $G$ by deleting a vertex in all possible ways (counted ac...

Conjecture For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two....

Conjecture If $G$ is a bridgeless cubic graph, then there exist 6 perfect matchings $M_1,\ldots,M_6$ of $G$ with the property that every edge of $G$ i...

Conjecture Every bridgeless graph has a nowhere-zero 5-flow....

Conjecture Every simple digraph of order $n$ with minimum outdegree at least $r$ has a cycle with length at most $\lceil n/r\rceil$...

Conjecture If a group has $r$ generators and exponent $n$, is it necessarily finite?...

Conjecture Every finite group is the Galois group of some finite algebraic extension of $\mathbb Q$....

Conjecture There exists a finite lattice that is not an interval in the subgroup lattice of a finite group....

Conjecture There is no odd perfect number....

Conjecture There exist infinitely many positive integers $n$ so that both $n$ and $n+2$ are prime....

Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely...

Conjecture Let $E/K$ be an elliptic curve over a number field $K$. Then the order of the zeros of its $L$-function, $L(E, s)$, at $s = 1$ is the Morde...

Let $B \subseteq {\mathbb N}$. The representation function $r_B: {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j...

Conjecture Every even integer greater than 2 is the sum of two primes....

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the co...

Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, then the extension field...

Conjecture For any $\epsilon>0$ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$ Since $\epsilon$ can be replaced by a smaller ...

Conjecture A Mersenne prime is a Mersenne number $$ M_n = 2^p - 1 $$ that is prime. Are there infinite number of Mersenne Primes?...

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 ...

Problem Is P = NP?...

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

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...

Conjecture Let $f\in Diff^{r}(M)$ and $p\in\omega_{f}$. Then for any neighborhood $V_{f}\subset Diff^{r}(M)$ there is $g\in V_{f}$ such that $p$ is pe...

Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected, compact and boundaryles manifold M and $\chi^{r}(M)$ the space ...