Unsolved Problems

Does every convex, closed, twice-differentiable surface in $\mathbb{R}^3$ have at least two umbilical points?...

Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?...

Does the Euler characteristic of a compact affine manifold vanish?...

What minimal hypersurfaces in spheres have constant mean curvature?...

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

Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?...

What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?...

Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?...

Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?...

Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?...

What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?...

What is the largest minimum area of a triangle determined by $n$ points in a unit square?...

Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...

What is the maximum number of unit distances determined by $n$ points in the plane?...

Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?...

Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...

Can all regular languages be expressed with generalized regular expressions of bounded star height?...

For which number fields is there an algorithm to determine solvability of Diophantine equations?...

Does the central limit theorem hold for all φ-mixing sequences?...

Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...

What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?...

Is the sphere the worst-packing convex solid?...

What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?...

Is there a constant $c > 1$ such that all non-cyclotomic polynomials have Mahler measure at least $c$?...

Is a measurable set in $\mathbb{R}^d$ spectral if and only if it tiles by translation?...

What is the maximum size of a cap set in $\mathbb{F}_3^n$?...

Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?...

What is the exact value of the Ramsey number $R(5,5)$?...

If a billiard table is strictly convex and integrable, is it necessarily an ellipse?...

How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?...

Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?...

For any $\varepsilon > 0$, is there a constant $c(\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\varepsilon) x^{1/2-\varepsilon}$?...

If Euler's totient function $\phi(n)$ divides $n-1$, must $n$ be prime?...

Does there exist a 3×3 magic square composed entirely of distinct perfect squares?...

Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?...

Does the partition function satisfy any arbitrary congruence infinitely often?...

Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?...

Are there infinitely many perfect numbers?...

Do quasiperfect numbers exist?...

Do any almost perfect numbers exist that are not powers of 2?...

Are there exactly 65 idoneal numbers, or could there be 66 or 67?...

Do any pairs of amicable numbers exist where one is odd and one is even?...

Are there infinitely many pairs of amicable numbers?...

Are there infinitely many Giuga numbers?...

Do Lychrel numbers exist in base 10?...

Do any odd weird numbers exist?...

Is $\pi$ a normal number in base 10?...

Are all irrational algebraic numbers normal in every base?...

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

If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?...