Unsolved Problems

Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For...

What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?...

Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs....

For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$....

Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?...

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1....

What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?...

For any $n$ runners on a circular track with distinct constant speeds, each runner is "lonely" (distance at least $1/n$ from all others) at some time....

Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\{0, 1, \ldots, |E|\}$ such that edge labels (absolute difference...

What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?...

A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal....

If a function on $\mathbb{R}^n$ has zero integral over every congruent copy of a given domain, must the function be identically zero?...

The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$....

What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?...

For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets....

Find all integer solutions to $n! + 1 = m^2$....

For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z....

Can every non-negative rational function be expressed as a sum of squares of rational functions?...

Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrang...

If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f...

If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?...

Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?...

What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?...

Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?...

For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in po...

For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$?...

Can every graph be totally colored with at most $\Delta + 2$ colors, where $\Delta$ is the maximum degree?...

Are there infinitely many Leinster groups?...

What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?...

Do Lychrel numbers exist in base 10?...

Is 10 a solitary number (no other number shares its abundancy index)?...

Does every nonnegative integer appear in Recamán's sequence?...

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

What is the maximum number of points in an $n \times n$ grid with no three collinear?...

Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?...

What is the outcome of a perfectly played game of chess?...

What is the perfect value of komi (compensation points) in Go?...

Are the nim-sequences of all finite octal games eventually periodic?...

Is the nim-sequence of Grundy's game eventually periodic?...

What is the optimal strategy for two agents to meet on a network without communication?...

For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?...

What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?...

What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?...

Can three unknotted space curves (not all circles) be arranged as Borromean rings?...

Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?...

If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors?...

What is the maximum chromatic number of biplanar graphs?...

Does every thrackle have at most as many edges as vertices?...

What is the maximum pathwidth of an n-vertex cubic graph?...

What is the longest induced path in an n-dimensional hypercube graph?...