Problems
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VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Is the closest vector problem NP-hard to approximate within polynomial factors n^c? Yes for some c > 0: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational…
In the circle of Kummer's regular primes and Vandiver's conjecture, the paper proves that almost all primes are partially regular, yielding a partial Vandiver theorem for a density-one set of primes, with consequences for Kubota-Leopoldt…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Grothendieck asked whether every finite locally free group scheme of order n is killed by n (its n-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne…
For triangular arrays of nodes a_i^n∈[-1,1] let L^nf be the Lagrange interpolation polynomials of a continuous f, with fundamental polynomials p_i^n. Is there a choice of nodes such that for every continuous f there is some x where…
If f(n) is the maximum total side length of n interior-disjoint squares packed in the unit square, is f(k^2 + 1) = k? An exact rational configuration packs 17 squares with total side length greater than 4, refuting the identity at k = 4.
Huang, Jiang and Oblomkov conjectured that the Eulerian q-series counting commuting pairs of nilpotent matrices with X^a = Y^b equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered…
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge…
Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its L^2 space? No. The Cantor measure with base b admits no Fourier frame for any odd integer b > 1,…
If Sidon sets A, B ⊆ 1, …, N satisfy (A-A) ∩ (B-B) = 0, must binom|A|2 + binom|B|2 ≤ binomf(N)2 + O(1), where f(N) is the largest Sidon-set size in [N] - and can the bound be improved by a fixed proportion when |A| = |B|?
Wegner conjectured in 1965 that every finite family R of axis-parallel rectangles satisfies τ(R) ≤ 2ν(R) - 1, where τ is the minimum number of piercing points and ν the largest pairwise-disjoint subfamily. False, by an explicit…
Litvak conjectured in 2018 that for every p > 0 the quantity E[min_i ≤ n |g_i|^p], for g ~ N(0,Σ), is minimized over n × n correlation matrices by the Gram matrix of the regular simplex in R^n-1. False: the matrix Σ^cos_ij = cos(π(i-j)/n)…
Let p be a complex polynomial of degree n ≥ 2 whose zeros all lie in the closed unit disk. Then for every zero a of p, there exists a critical point ζ of p such that |ζ-a| ≤ 1. This is the standard Sendov statement and exactly matches the…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.
For the switch-walk-switch walk on Z_2 wr Z started at (0,0) and (0,2), prove |P_t^x - P_t^y|_TV asymp t^-1/2.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.