Unsolved Problems

For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?...

If R is a regular local ring and P,Q are prime ideals with $\dim(R/P) + \dim(R/Q) = \dim(R)$, is $\chi(R/P, R/Q) > 0$?...

Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?...

Is the assembly map in K-theory an isomorphism for all locally compact groups?...

Are Berge knots the only knots in S³ admitting lens space surgeries?...

Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...

If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?...

Are certain polynomials in Pontryagin classes homotopy invariants?...

Can unknots be recognized in polynomial time?...

Do quantum invariants of knots determine their hyperbolic volume?...

Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...

Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...

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

If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...

Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...

For any finite union-closed family of sets, does some element appear in at least half the sets?...

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

Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?...

Do any odd perfect numbers exist?...

Are there infinitely many perfect numbers?...

Do quasiperfect numbers exist?...

Do Lychrel numbers exist in base 10?...

Do odd weird numbers exist?...

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

Is π a normal number (all digits equally frequent in all bases)?...

Are all irrational algebraic numbers normal?...

Does iterating unsigned differences on prime sequence always yield 1 as first element?...

If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?...

Are there infinitely many real quadratic fields with class number 1 (unique factorization)?...

Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields....

Does the p-adic regulator of an algebraic number field never vanish?...

Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?...

For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?...

Is the Euler-Mascheroni constant γ irrational? Transcendental?...

For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?...

If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?...

Can integer factorization be done in polynomial time?...

Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?...

What is the optimal sphere packing density in dimensions >3?...

Let $a_1, \ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \ldots, n + a_k$ are all prime, p...

For all integers $x, y \geq 2$, we have $\pi(x+y) \leq \pi(x) + \pi(y)$, where $\pi(n)$ denotes the prime counting function (the number of primes less...

For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\gcd(a,b,c) = 1$, and discriminant $\Delta = b^2 - 4ac$ not a perfect square, the polynomial ta...

Let $\pi(n; a, b)$ be the number of primes $p \le n$ with $p \equiv a \pmod b$. For every $a$ and $b$ with $a \perp b$, are there infinitely many valu...

Let $\{a_i\}$ be any infinite sequence of integers for which $\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progress...

Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes...

Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime....

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

Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \ln n \ln \ln n \ln \ln \ln \ln n / (\ln \ln \ln n)^2$ for arbitrarily large constan...

Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?...

For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?...