The Erdős minimum overlap problem
Pin down the constant c ≈ 0.38 such that every split of {1,…,2N} into equal halves has a difference occurring cN times.
问题
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符合的问题
Spherical designs with few points
Find spherical t-designs on the 2-sphere with fewer points than known and prove they exist; t = 19 and t = 21 are the frontier.
Spreading points evenly: the max-to-min distance ratio
Place n points to minimize the ratio of the largest to the smallest distance; exact values in the plane are known only up to n = 8.
Arbitrarily long runs of equal signs of the Liouville function
Prove that the Liouville function takes the same sign on arbitrarily long blocks of consecutive integers.
The lonely runner conjecture
Find integer speeds for which no moment leaves every runner at least 1/(n+1) from the start, or prove the conjecture, now proved for n ≤ 14.
The Littlewood conjecture
Prove that n‖nα‖‖nβ‖ gets arbitrarily close to 0 for every pair of real numbers α and β.
Lipschitz graphs without three-term progressions
Find a function N → Z with bounded steps whose graph has no three-term progression, or prove none exists (the additive square problem).
Covering a polynomial lemniscate by discs
Show that the set where a monic polynomial has modulus at most 1 can be covered by discs whose radii sum to at most 2.
Lehmer's totient problem
Decide whether some composite number n has Euler's totient φ(n) dividing n − 1, as every prime does.
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.
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.
Imaginary quadratic fields whose class groups have large ℓ-rank
Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.
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.
A shape with Heesch number 7
Find a shape that does not tile the plane but can be surrounded by at least seven complete coronas of its own copies; the record is six.
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.
Hall's conjecture: a square extremely close to a cube
Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.
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.