Unsolved Problems

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 any odd weird numbers exist?...

If $B$ is an additive basis of order 2, must the representation function tend to infinity?...

If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \geq k$?...

Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?...

Can an algorithm determine if a constant-recursive sequence contains a zero?...

Do the Ulam numbers have a positive density?...

Is there a dense set of points in the plane with all pairwise distances rational?...

Can integer factorization be solved in polynomial time on a classical computer?...

Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?...

Is $p$ prime if and only if $pB_{p-1} \equiv -1 \pmod{p}$ for the Bernoulli number $B_{p-1}$?...

Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?...

Do symmetric informationally complete POVMs exist in all dimensions?...

Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?...

Can a finite system of left cosets forming a partition of a group have distinct indices?...

Is there a minimum positive Mahler measure for non-cyclotomic polynomials?...

For which domains do non-zero functions exist with zero integrals over all congruent copies?...

Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?...

If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time?...

For a finite family of sets closed under unions, must some element appear in at least half the sets?...

For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?...

Does every bridgeless graph have a collection of cycles covering each edge exactly twice?...

Does every finite connected vertex-transitive graph have a Hamiltonian path?...

What is the chromatic number of the plane with unit distance graph coloring?...

Can unknots be recognized in polynomial time?...

Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...

Does the partition principle (PP) imply the axiom of choice (AC)?...

Does the generalized continuum hypothesis imply the existence of an $\aleph_2$-Suslin tree?...

Does there exist a Jónsson algebra on $\aleph_\omega$?...

Is the open coloring axiom (OCA) consistent with $2^{\aleph_0} > \aleph_2$?...

What is the largest possible cap set in $n$-dimensional affine space over the three-element field?...

Does the Ibragimov-Iosifescu conjecture hold for φ-mixing sequences?...

What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, ...

Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?...

Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?...

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