Mathematics Problem Archive

Jaeger: If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y in F^(n) which haveal...

Graham: Is x^(2)+y^(2)=z^(2) partition regular? That is, is it true that every coloring of the positive integers by a finite number of colors contains...

Erdős-Strauss : Is it possible, for each positive integer n, to find positive integers a, b, and c so that 4/n = 1/a + 1/b + 1/c ? See this....

Singmaster : Show that there is some B so that no integer appears more than B times among the binomial coefficients. See this....

Carmichael : There is no n so that the only integer m with phi(n) = phi(m) is m=n. ("phi" is the Euler phi/totient function). See this....

Ulam : Is there a dense of points in the real plane so that every two points are at a rational distance? See this....

A covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one of the codewords by changing at most R bits...

Chung/: A de Bruijn covering code of radius R is a binary string so that the set of words appearing as n consecutive symbols (with wrap-around) is a c...

Kolakoski: There is (essentially) a unique sequence over {1,2} which is its own run-length encoding. Is the density of 1's in this sequence 1/2? See t...

Alon: What is the threshold function n = f(k) for the event that a random permutation on n symbols contains all patterns on k symbols? Conjecture: f(k...

Tao: What is the probability that a random nXn matrix over Z_(p) has zero permanent as n goes to infinity? (Surely 1/p... as long as p is not 2.)...

Is the exponent of matrix multiplication 2? In other words, can two n b n matrices be multiplied in O(n^(2+)^(ľ)) steps? See this....

Let $A=(a_{ij})$ be an $n\times n$ matrix with nonnegative entries and total entry sum $n$. Define $$\phi(A)=\prod_{i=1}^n\sum_{j=1}^n a_{ij}+\prod_{j...

A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possibl...

Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Fix singularities $a_1,\ldots,a_n$ and real exponent differences $\alpha_1,\ldots,\alpha_n$ for second-order Fuchsian equations on the Riemann sphere....

For the Heun equation $$y''+\left(\sum_{j=0}^2\frac{1-\alpha_j}{z-a_j}\right)y'+\frac{Az-\lambda}{(z-a_0)(z-a_1)(z-a_2)}y=0,$$ where $\alpha_j>0$, $A=...

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be rational with Julia set $J$, and suppose that a component $D$ of $\widehat{\mathbb C}\setminus J$...

How many completely invariant components can the Fatou set of a transcendental entire function have? In particular, can there be more than one?...

Does there exist a rational function having an analytic invariant Jordan curve on which it is topologically conjugate to an irrational rotation, where...

Let $C$ be an analytic invariant curve of a rational function $f$, suppose $f:C\to C$ is not a homeomorphism, and assume $C$ contains a repelling fixe...

Let $D\subset\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzer...

A spherical quadrilateral is a disk with four marked boundary vertices, curvature-one metric, geodesic sides, and interior angles $\pi\alpha_j>0$. Cla...

For a local inverse germ $\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along alm...

Let $F$ be entire in two variables and let a holomorphic germ $\phi$ satisfy $F(z,\phi(z))=0$ near a nonsingular point. Must $\phi$ admit analytic con...

Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\{u_k<0\}$ and $D_k^*$ is the component containing $-1/2$...

Let $u$ be subharmonic on $1<|z|<2$, and let pairwise disjoint open sets $D_k$ be unions of components of $\{u<0\}$. Suppose that for every $r\in(1,2)...

Let positive masses $a_k$ be placed at a discrete set $x_k\in\mathbb R^n$ and suppose $\sum_k a_k/|x_k|^{n-1}<\infty$, so that $$F(x)=\sum_k\frac{a_k(...

Let $f$ be holomorphic in the unit disk with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counted with multiplicity. Determine the larg...

Let $f$ be holomorphic in the unit disk, with $f(0)=0$, $f'(0)\ne0$, no other zeros, and exactly two solutions $z_1,z_2$ of $f(z)=1$, counted with mul...

Solve the two-one-point extremal problem for real holomorphic $f$: determine the minimum of $\max(|z_1|,|z_2|)$ and maximum of $|f'(0)|$ when $f(0)=0$...

Let $f$ be real and holomorphic in the unit disk, with one simple zero at $0$ and two simple $1$-points at $\pm ia$. Determine the minimum possible va...

Obtain better rigorous estimates for the Littlewood exponents $\alpha$, $\beta$, and for $\sup_c P_c$, where $P_c$ is the pressure for the hyperbolic ...

Are the iterates $p_c^n$ of hyperbolic quadratic polynomials extremal, or nearly extremal, for the Littlewood exponent $\alpha$ governing mean spheric...

For which parameters $c$ is the pressure $P_c$ of the hyperbolic quadratic polynomial $p_c(z)=z^2+c$, for the potential $|(p_c^n)'|^{-1}$, close to or...

For the rectangular lattice $\Lambda=\{an+ibm:n,m\in\mathbb Z\}$ with $a^2+b^2=1$ and $a\in(0,1)$, let $f_a:\mathbb D\to\mathbb C\setminus\Lambda$ be ...

Let $f=(f_0,\ldots,f_n)$ be a linearly nondegenerate holomorphic curve, let $T(r,f)$ have finite lower order $\lambda$, and let $N_1(r)$ be the averag...

Let $f$ be holomorphic in the unit disk with a simple zero at $0$, exactly two simple $1$-points at $a$ and $\overline a$, and no other zeros or $1$-p...

Given $n$ points and $m$ hyperplanes in $\mathbb R^d$ whose incidence graph contains no $K_{s,t}$, determine the maximum number of incidences. Of spec...

For an $n$-point planar set, determine the maximum number of halving lines. More generally, determine the maximum number of $k$-sets, subsets obtained...

For $n$ pseudocircles in general position, determine the maximum number of tangent pairs and the maximum number of digon cells. Determine whether the ...

Determine the worst-case complexity of the medial surface and of an offset surface of an $n$-feature polyhedron, and of the Voronoi diagram of $n$ lin...

Determine the exact Erdős–Szekeres number $f(n)$, the least number of planar points in general position forcing a convex $n$-gon. Also determine sharp...

Determine the combinatorial complexity of the visibility complex of $n$ pairwise disjoint unit spheres in three-dimensional space....

Given $n$ planar points, find a subquadratic algorithm for the minimum-area triangle or prove a quadratic lower bound in a suitable computation model....

Given $n$ points in $\mathbb C^2$, determine in quadratic time whether three lie on a complex line, or prove a quadratic lower bound; the known algori...

Determine the optimal complexity of computing the volume of the union of axis-aligned boxes in fixed dimension at least three. In particular, is there...

Given a planar point set $P$, sample uniformly from the simple polygons with vertex set $P$ in polynomial time, or determine the complexity of countin...