Mathematics Problem Archive

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

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 quasiperfect numbers exist?...

Do odd weird numbers exist?...

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

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?...

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....

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?...

Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity?...

(Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?...

(Eilenberg-Ganea) Is there a group G of cohomological dimension 2 and geometric dimension 3?...

Does every Artin group have a finite $K(G,1)$?...

(Andrews-Curtis) If K and L are simple homotopy equivalent finite 2-complexes, can one transform K to L by a sequence of elementary collapses and expa...

(Belegredek) Is there a 3-complex X (not necessarily aspherical) which is not homotopy equivalent to a 2-complex but H³(X; {G}) = 0 for all local coef...

Study the quasi-isometry group QI(Rn). How big is it?...

(Bridson) Is G×Z quasi-isometric to G for G =Thompson’s group? Any f.g. group?...

(Lubotzky) Does Out(Fn) have the congruence subgroup property?...

(Yves de Cornulier) Let G be residually torsion-free nilpotent. Is G Haagerup (= a-(T)-menable)?...

(Henry Glover) Does for every finite graph G the following 1−2−∞- conjecture hold: G is planar, a double cover of G is planar, or no finite cover is p...

Let $L$ be a flag triangulation of $S^{2k-1}$, let $f_i$ be the number of $i$-simplices of $L$, and define $$\chi=1-\sum_{i=0}^{2k-1}(-1)^i\frac{f_i}{...

Do there exist groups G with balanced presentation (same number of generators and relations), with H1(G) = 0 and with unsolvable word problem?...

Is there a sequence of (perfect, of course) groups with balanced presentations among which one cannot recognize trivial groups?...

Let $G$ be a closed transitive subgroup of the automorphism group of a rooted tree. Must every level stabilizer of $G$ contain an element acting witho...

Prove the conjectured local-to-global principle for linear ordinary differential equations known as the Grothendieck–Katz $p$-curvature conjecture....

Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?...

Study geometric properties of Markov spectrum....

Determine when the minimizer $y^*$ in Theorem 2.1 of the source is smooth....

With the terminology of Problem 2.20, denote by $\mathcal{F}(f)$ the set of points where the sequence $\{f_n(z)\}$ is not normal. Fatou asks if there ...

Let $B$ be the Bergman space of square integrable functions in $\mathbb{D}$, that is, those functions $f(z) = \sum^\infty_{n=0} a_nz^n$ for which $\su...

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$ with at least two boundary points, and let the function $\phi$ map $\Omega$ analytically and...

For $-\infty<p<+\infty$ let \[B(p):=\sup\{\beta_f(p):f\text{ conformal map of }\mathbb{D}\text{ into }\mathbb{D}\}\] where \[\beta_f(p)=\limsup_{r\to1...

In their famous Acta paper , Hardy and Littlewood introduced the celebrated Hardy-Littlewood maximal function in connection with complex function theo...

For an iteration without memory that uses $n$ evaluations of a function or its derivatives per step, is its convergence order always at most $2^{n-1}$...

Given a complex polynomial $f$ of degree $d\geq2$ and $z\in\mathbb{C}$, must there be a critical point $c$ of $f$ such that $|f(z)-f(c)|\leq |f'(z)|\,...

Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? ...

How many mutually unbiased bases are there in 6 dimensions?...

What is the value of the (real) Grothendieck constant?...

3n+1 ("Collatz" or "Ulam") problem: Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three a...

Are the positive integer powers of 3/2 mod 1 uniformly distributed in the unit interval? One would think so, but apparently this is a hard question. S...