Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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For any natural number n, let Δ(n) be the size of the smallest set B of integers such that every natural number from 1 to n is expressible as a difference of two elements of B (such sets are known as difference bases for the interval 1,…,n). Write C(n) := Δ^2(n)/n, and C := inf_n ≥ 1 C(n).
Let d(n) be the divisor function. The Dirichlet divisor problem concerns the error term Δ(x) := Σ_n≤ x d(n) - x(log x + 2γ -1), where γ is Euler's constant. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a> Define the divisor-problem exponent α := infBigla≥ 0: Δ(x)=O(x^a+ε) for all ε>0Bigr.
Let Λ denote the von Mangoldt function. For coprime positive integers a,q, define ψ(x;q,a) := Σ_n≤ x, n≡ a (mod q) Λ(n).
Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?
For any transcendental entire function f(z)=Σ_n≥ 0 a_n z^n, define . M(r,f):=max_|z|=r|f(z)|, μ(r,f):=max_n≥ 0|a_n| r^n. Following [Er1961], define β(f):=liminf_r→∞μ(r,f)/M(r,f). We define C_51 = B to be the supremum of β(f) over all transcendental entire functions f.
For any natural number N, let C(N) denote the largest cardinality of a subset A of 1,…,N with the property that ab+1 is square-free for all a,b ∈ A. Establish upper and lower bounds for C(N) that are as strong as possible.
For any natural n, let C(n) denote the maximum possible sum of side lengths of n squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for C(n) that are as strong as possible.
Let G be a finite graph with minimum degree at least 3. Must G contain a cycle of length 2^k for some k ≥ 2?
A family of three distinct sets A,B,C is a 3-sunflower (or Δ-system) if A∩ B = A∩ C = B∩ C. A family of sets is sunflower-free if it contains no 3-sunflower (equivalently, no sunflower of any size ≥ 3). Let [n]:=\1,2,…,n\ and let f(n) denote the maximum size of a sunflower-free family F⊆ 2^[n].
Let overlineℚ be the set of all algebraic numbers. The naïve height h : overlineℚ → ℝ is defined as follows. Let α ∈ overlineℚ and let P(x) be an irreducible primitive polynomial with integers coefficients such that P(α)=0. Let n be the degree and a be the leading coefficient of P(x).
Let p_n denote the n-th prime and, for m ≥ 1, write H_m := liminf_n → ∞ (p_n+m - p_n) for the least limit point of the gaps between primes m apart.
Let H be an infinite-dimensional complex Hilbert space and let B(H) be the Banach algebra of bounded operators on H, equipped with the operator norm.
C_17 is the limit (if it exists) of R(k)^1/k as k → ∞, where the diagonal Ramsey number R(k) is the smallest integer n such that every red/blue colouring of the edges of the complete graph K_n contains a monochromatic copy of K_k.
For a natural number N, let C(N) be the largest quantity such that N! can be factored into N factors that are greater than or equal to C(N) (see OEIS A034258). Establish upper and lower bounds on C(N) that are as strong as possible.
The Falconer distance problem threshold C_34 = s_Δ(ℝ^2) in the plane is defined as s_Δ(ℝ^2) : :=\ infBigl s∈[0,2] : ∀ compact E⊂ℝ^2,\ dim_H(E)>s Longrightarrow lvertΔ(E)rvert>0 Bigr.
Let E⊂ ℝ^2 be a planar set. The Favard length of E is defined by Fav(E) := 1/π∫_0^π lvert Proj R_θ Ervert dθ, where Proj is orthogonal projection to the horizontal axis and R_θ is rotation by angle θ.
A convex body K⊂ℝ^d is hollow (lattice-free) with respect to a lattice Λ if int(K)∩Λ=∅. <a href="#CS2019-hollow-def">[CS2019-hollow-def]</a> For a hollow body, the lattice width is w(K) := min_u∈ℤ^d∖0 (max_x∈ Ku· x-min_x∈ Ku· x).
Let r(N) be the maximum size of a subset A⊂\1,…,N\ with no non-zero square differences a-b=n^2.Then C_4b is the least constant such that r(N) ≤ N^C_4b+o(1).
Let 1 ≤ q ≤ ∞, and let j and m be non-negative integers such that j < m. Furthermore, let 1 ≤ r ≤ ∞, p ≥ 1 be real and θ ∈ [0, 1] such that the following relations hold: 1/p = j + θ ( 1/r - m ) + 1 - θ/q, j/m ≤ θ < 1.
Let N(t) := \#(m,n)∈ℤ^2: m^2+n^2≤ t^2 be the number of integer lattice points inside the (closed) disk of radius t centered at the origin. The Gauss circle problem is to find the smallest exponent θ such that, for every ε>0, N(t) = π t^2 + O(t^θ+ε).
C_43 is defined as the infimum of the ratio of the length of the Steiner Minimal Tree to the length of the Euclidean Minimum Spanning Tree over all finite sets of points V ⊆ ℝ^2: C_43 = inf_VL_S(V)/L_M(V), where L_S(V) and L_M(V) denote the lengths of Steiner Minimal Tree and Minimum Spanning Tree…
We define C_56 = δ_2 to be the smallest real number δ ≥ 0 such that the following uniform bound toward the Generalized Ramanujan Conjecture holds.
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
If n,m are natural numbers, let C(n,m) denote the maximum number of incidences that are possible between n points and m lines in the plane. Establish upper and lower bounds on C(n,m) that are as strong as possible.
C_39=H_3 is the Hadwiger covering number in dimension 3, which can also be formulated in terms of illumination of the boundary.
Let C denote the best constant for which | x: sup_h>0 1/2h ∫_x-h^x+h f(y) dy ≥ λ | ≤ C/λ ∫_ℝ f(x) dx for absolutely integrable non-negative f : ℝ → ℝ. What is C?
For 1 ≤ p ≤ 2, let C(p) be the best constant such that ‖ hat f ‖_L^p'(ℝ) ≤ C(p) ‖ f ‖_L^p(ℝ) holds for all test functions f : ℝ → ℝ. Here p' := p/p-1 is the dual exponent of p. What is C(p)?
For any n ≥ 3 and any convex body K in the plane, let C(n,K) be the largest quantity such that in every configuration of n points in K, there exists a triple of points determining a triangle of area at most C(n,K) times the area of K. Establish upper and lower bounds on C(n,K).
For any n ≥ 3 let C(n) be the largest quantity such that in every configuration of n points in the plane, there exists a triple of points determining a triangle of area at most C(n) times the area of their convex hull. Establish upper and lower bounds on C(n).
Let C be the best constant for which one has ‖f f‖_L^2(ℝ)^2 ≤ C ‖ff‖_L^1(ℝ) ‖f * f‖_L^∞(ℝ) for non-negative f : ℝ → ℝ. What is C?
C_33=A(2) is the Ihara constant over 𝔽_2. <a href="#DM2013-def-Aq">[DM2013-def-Aq]</a> For each integer g≥ 1, let N_2(g) := maxbigl\#X(𝔽_2) : X/𝔽_2 a smooth projective geometrically integral curve of genus gbigr. <a href="#DM2013-def-Nqg">[DM2013-def-Nqg]</a> Then A(2) := limsup_g→inftyN_2(g)/g.
Let G = (g_ij) be an M × N random matrix with independent standard Gaussian entries, and let Z(G) := | σ ∈ -1,1^N : G σ ≥ 0 coordinatewise |. This is the zero-margin binary (or Ising) perceptron. Write M = ⌊ α N ⌋. Define C_80 to be the infimum of all α > 0 such that ℙ(Z(G) > 0) → 0 as N → ∞.
Let d ≥ 1, and let q be a prime power. Let 𝔽_q be a finite field of order q. A Kakeya set is a set K that contains a line in every direction, and an Nikodym set N is a set with the property that every point x in 𝔽_q^d is contained in a line that is contained in N ∪ x.
Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…
C_3b = SD(\0,1,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|)^C_3b whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± B := a ± b: a ∈ A, b ∈ B.
C_24 is the Komlós discrepancy constant (often denoted K). For a real matrix A∈ℝ^m× n, define its (sign) discrepancy by disc(A) := min_x∈-1,1^n ‖Ax‖_∞. For each n≥ 1, define the dimension-n Komlós discrepancy K_n := supdisc(A): A∈ℝ^n× n and ‖A_ast j‖_2≤ 1 for all columns j.
Let D:=\z∈ℂ:lvert zrvert<1\. The Bergman space A^2(D) consists of analytic functions f on D with lVert frVert_2 := (1/π∫_D lvert f(z)rvert^2 dA(z))^1/2 < ∞, where dA(z) denotes the Lebesgue area measure. For c∈(0,1), write A(c,1) := z∈ℂ: c<lvert zrvert<1.
Let D=\z∈ℂ:lvert zrvert<1\ and let F be the class of holomorphic functions f:D→ℂ normalized by f'(0)=1. <a href="#BS2023-def-F">[BS2023-def-F]</a> For finF, let L_f denote the radius of the largest disk contained in f(D).
C_13b = a is the infimal area of a convex planar set Ω that can cover a congruent copy of every convex planar set of diameter 1.
Let f(x)=Σ_i=0^n a_i x^i = a_nΠ_i=1^n (x-α_i) be a polynomial with complex coefficients. The Mahler measure of f is M(f) := |a_n|Π_i=1^n max1,|α_i|.
Define the infimal exponent μ_ζ by μ_ζ := infBiglθ≥ 0: lvertζ(1/2+it)rvert≪_ε(1+lvert trvert)^θ+ε for all ε>0Bigr. We define C_62a := μ_ζ, the Lindelof (pointwise growth) exponent for ζ(1/2+it).
For integers q≥ 2 and a with gcd(a,q)=1, let P(a,q) denote the least prime in the arithmetic progression a bmod q. <a href="#Xyl2011-def-Paq">[Xyl2011-def-Paq]</a> Linnik's theorem asserts that there exist constants C,L>0 such that P(a,q) ≤ C q^L (gcd(a,q)=1), uniformly for all q≥ 2.
Let K⊂ℝ^n be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying K=-K. Its polar body is K^∘ := y∈ℝ^n: ⟨ x,y⟩ ≤ 1 for all x∈ K. The volume product of K is vp(K) := Vol_n(K) Vol_n(K^∘).
For a number field K, let Δ_K denote the absolute value of its discriminant and let [K:ℚ] denote its degree. The root discriminant of K is rd(K) := Δ_K^1/[K:ℚ].
C_18 is the least constant such that, whenever A is a subset of 𝔽_2^n with lvert A+Arvert ≤ Klvert Arvert, then A can be covered by K^C_18+o(1) cosets of a subspace of cardinality at most lvert Arvert, where the limit o(1) is with respect to the limit K → ∞.
For positive-semidefinite d × d matrices A_1, …, A_n and any unitarily invariant norm |||·||| (including the operator norm and Schatten p-norms) and m ≤ n, define C(n,m,d) := inf frac 1/n^m Σ_j_1, j_2, …, j_m = 1^n |||A_j_1A_j_2… A_j_m||| (n-m)!/n! Σ_substackj_1, j_2, …, j_m = 1 \ all distinct^n…
Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).
C_27b is the highest possible chromatic number for any biplanar graph.
For 0 ≤ ρ ≤ 1, let C(ρ) denote the largest quantity such that any graph on n vertices and (ρ+o(1)) C(n, 2) edges will have at least (C(ρ)-o(1)) C(n, 3) triangles. What is C(ρ)?
C_13a is the infimal area of a convex domain Ω that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 1.
