Lehmer's Mahler measure problem
Beat Lehmer's 1.17628 with an integer polynomial of smaller Mahler measure above 1, or prove the measures stay away from 1.
问题
开放问题和研究问题，每个都是一个 git 仓库：很多来自精选合集，并注明出处。路线是任何人都能开启或接续的分支；检验通过后，结果才算数。
符合的问题
Leech trees beyond six vertices
Find an edge-weighted tree on more than six vertices whose pairwise path lengths are exactly 1, 2, …, n(n−1)/2, or prove that none exists.
Lebesgue's universal covering problem
Find the least area of a convex shape that covers every planar set of diameter 1, now known to lie between 0.832 and 0.8440936.
Are the largest roots of a random polynomial real?
Find the limiting probability that the largest roots of a random polynomial with Gaussian coefficients are all real.
Density of ones in the Kolakoski sequence
Prove that the self-describing Kolakoski sequence of 1s and 2s has a density of 1s, equal to 1/2.
Kissing numbers
Improve the bounds on kissing numbers in dimensions up to 48, starting with dimension 5, where 40 ≤ τ(5) ≤ 44.
Open据 Kissing numbers and spherical codes (Henry Cohn)，2026-10-20
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Does the Jones polynomial detect the unknot?
Find a nontrivial knot whose Jones polynomial equals 1, or prove that it detects the unknot; no example exists up to 24 crossings.
The irrationality measure of pi
Lower the proven upper bound 7.10186 on the irrationality measure of π, which is conjectured to equal 2.
Is the group of interval exchange transformations amenable?
Decide whether the group of interval exchange transformations of the circle is amenable.
Integer distances in general position
Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
The inscribed square problem (Toeplitz's square peg problem)
Show that every simple closed curve in the plane passes through the four vertices of some square.
Imaginary quadratic fields with a prescribed class number
Give an algorithm that, for h between 2^90 and 2^100, quickly finds an imaginary quadratic field with class number exactly h.
Global regularity for 2D Burgers with hyper-dissipation
Prove global regularity, or blow-up, for the two-dimensional Burgers equation with bi-Laplacian dissipation and large initial data.
The Hirsch conjecture in dimension four
Find a 4-dimensional polytope with n facets whose graph has diameter greater than n − 4, or prove the Hirsch bound in dimension 4.
Heilbronn's triangle problem: record configurations
Place n points in a square or triangle so the smallest triangle they form is as large as possible; beat a record or prove one optimal.
The Erdős–Szekeres happy ending problem
Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
The Häggström–Pemantle conjecture on competing growth
Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.
Is the Gompf–Scharlemann–Thompson knot ribbon?
Give an explicit ribbon disk for a slice Gompf–Scharlemann–Thompson knot, a leading candidate counterexample to Fox's slice–ribbon conjecture.
The graceful tree conjecture
Show that every tree with m edges can be labelled by distinct numbers 0 to m so that the edge differences are exactly 1 to m.
Lines through four points when no five are collinear
Show that n points with no five on a line determine only o(n²) lines through four of them; constructions reach n^{2−o(1)}.
Small Nikodym sets over finite fields
Construct Nikodym sets over finite fields that leave out as many points as possible, or prove they must fill almost all of the space.
The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
The Erdős–Turán conjecture on additive bases
Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
The Erdős–Straus conjecture
Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
The Erdős similarity problem
Show that for every infinite set A of reals some set of positive measure contains no scaled and shifted copy of A.
Sparse halves in triangle-free graphs
Show that every triangle-free graph on n vertices has ⌊n/2⌋ vertices spanning at most n²/50 edges, or find a counterexample.
The Erdős matching conjecture
Determine the largest r-uniform hypergraph on n vertices with no k disjoint edges; Erdős conjectured one of two simple families wins.
The Erdős–Hajnal conjecture
Show that graphs avoiding a fixed induced subgraph H always contain a clique or independent set of size n^{c(H)}.
Cycles of length a power of two (the Erdős–Gyárfás conjecture)
Show that every finite graph with minimum degree 3 has a cycle of length 4, 8, 16 or another power of two, or find a counterexample.
The Erdős distinct distances problem
Show that n points in the plane always determine at least c·n/√(log n) distinct distances, as the square grid does.