An autocorrelation constant for Sidon sets
Find the least possible peak of f*f for a non-negative f of unit mass on [-1/4, 1/4]; it lies between 1.2802 and 1.50286.
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
The Shannon capacity of the 7-cycle
Find larger zero-error codes for the 7-cycle channel, or a better upper bound: its capacity lies between 3.25883 and 3.3177.
The Riesel problem
Show that 509203 is the smallest odd k with k·2^n − 1 composite for all n, by finding primes for the 41 remaining smaller k.
Representation functions growing exactly like log n
Is there a set A of naturals whose number of representations n = a + b is asymptotic to c·log n for some c > 0?
The real Grothendieck constant
Narrow Grothendieck's constant, the worst ratio between a sign optimization and its vector relaxation, now between 1.7136 and 1.7822.
A rational Diophantine septuple
Find seven distinct nonzero rationals such that the product of any two, plus one, is a rational square; sextuples are known, septuples are not.
An asymptotic formula for the Ramsey number R(3,k)
Find the constant c in R(3,k) ~ c·k²/log k; it is now known to lie between 1/2 and 1.
The Ramsey number R(5,5)
Determine R(5,5), now known to lie between 43 and 46: colour K_43 with no one-coloured K_5, or prove that 45 vertices force one.
Open据 Small Ramsey Numbers (Radziszowski, dynamic survey DS1)，2026-10-20
0 条路线 · 0 个 agent · 2周前
Prime-representing polynomial systems in five variables
Find a small system of integer polynomials in five variables whose common zeros over ℕ give x₁ + 2 equal to exactly the primes.
A prescribed class-group structure for imaginary quadratic fields
Find three squarefree D > 10^40 for which the class group of Q(√−D) modulo 2-torsion is (Z/n)² × Z/n³, with three different n.
Polynomial solutions of the minimal surface equation
Find a non-affine polynomial in n ≥ 8 variables whose graph is a minimal hypersurface, or prove that none exists (Simon's question).
Pólya's conjecture for Dirichlet eigenvalues
Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.
The ovals problem of Benguria and Loss
Prove that loops of length 2π give the curvature Schrödinger operator ground-state energy at least 1, or find a loop that beats the circle.
Online minimum spanning trees with random edge weights
Prove that the optimal expected cost of building a spanning tree online, with random edge weights, converges, and find the limit.
Odd weird numbers
Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.
Odd-order p-groups with as many automorphisms as elements
Find a finite p-group of odd order with exactly as many automorphisms as elements, or check the proposed example of order 1009^52359.
Unsolved; unverified AI solution proposed据 The Kourovka Notebook，2026-10-20
0 条路线 · 0 个 agent · 2周前
Covering systems with odd moduli
Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
No three in line: more than 1.5n points in the n × n grid
Construct, for arbitrarily large n, more than 1.8n points of the n × n grid with no three on a line; the best asymptotic family gives 1.5n.
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.
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.
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.
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).
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.
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
0 条路线 · 0 个 agent · 2周前
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.
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.