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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In harmonic analysis, for λ > 0 let T^λ denote the Bochner–Riesz operator on ℝ^3, initially defined for Schwartz functions f ∈ S(ℝ^3) by T^λ f(x) := ∫_ℝ^3 (1-lvert ξ rvert^2)_+^λ widehatf(ξ)e^ix· ξ dξ, where widehatf denotes the Fourier transform of f and (t)_+ := max\t,0\.
C_3c = SD(\0,1,2,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|, |A stackrelG+ 2B|)^C_3c whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± rB := a ± rb: a ∈ A, b ∈ B.
For any dimension n, let C(n) denote the quantity C(n) := π^n/2/Γ(n/2+ 1) inf_f (r/2)^n f(0)/hat f(0) where f ranges over integrable continuous functions f := ℝ^n → ℝ, not identically zero, with hat f(ξ) ≥ 0 for all ξ and f(x) ≤ 0 for all |x| ≥ r for some r>0.
C_5a is the smallest constant such that Sidon sets in \1,…,N\ have cardinality N^1/2 + (C_5a + o(1))N^1/4.
The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.
C_1a is the largest constant for which one has max_-1/2 ≤ t ≤ 1/2 ∫_ℝ f(t-x) f(x) dx ≥ C_1a (∫_-1/4^1/4 f(x) dx)^2 for all non-negative f : ℝ → ℝ.
Let C be the smallest constant such that min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C ‖f‖_L^1(ℝ)^2 for f ∈ L^1(ℝ). What is C?
Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). What is C?
Quantum Max Cut is the quantum analog of Max Cut. Given a graph G = (E,V), it asks for the maximum eigenvalue of H_G = Σ_(ij) ∈ E (I - X_iX_j - Y_iY_j - Z_i Z_j), where X_i, Y_i, Z_i are the Pauli matrices acting on the i'th tensor factor and trivially on all other coordinates.
For integers n ≥ 2 and t ≥ 1, an n × t partial Hadamard matrix is a matrix with entries in \± 1\ whose rows are pairwise orthogonal. Let N_n,t denote the number of such matrices. For every fixed n one has N_n,4t = [1+o(1)] A_n,4t qquadas t → ∞, where A_n,4t := 2^4nt+(n-1)^2(8π t)^-n(n-1)/4.
Let α be a nonzero algebraic number of degree d, with minimal polynomial over ℤ f(X)=a_dΠ_i=1^d (X-α_i), where a_d>0 and α_1,…,α_d are the conjugates of α. Define the Mahler measure of α by M(α) := a_dΠ_i=1^d max1,lvert α_irvert.
For each integer n ≥ 1, let S_n be the symmetric group on n letters. Over a base field k, the essential dimension ed_k(S_n) is the smallest integer d such that the general degree-n polynomial x^n + a_1 x^n-1 + ⋯ + a_n can be reduced to a d-parameter form by a Tschirnhaus transformation.
For each integer d ≥ 3, let ℓ_0(d) denote the maximal number of lines contained in a smooth surface of degree d in ℙ^3_ℂ. We define C_76 := limsup_d→∞ℓ_0(d)/d^2. The constant C_76 measures the quadratic growth rate of the maximal line count on smooth complex degree-d surfaces.
In harmonic analysis, the Beurling–Ahlfors transform B (also called the Ahlfors–Beurling operator) is the singular integral operator on L^p(ℂ), 1<p<∞, defined by Bf(z) = -1/π p.v.∫_ℂ f(w)/(z-w)^2 dm(w) = -1/π lim_ε→ 0^+∫_lvert w-zrvert>varepsilonf(w)/(z-w)^2 dm(w), where dm is Lebesgue measure on…
Let D=\z∈ℂ:lvert zrvert<1\. Following standard notation, let F be the class of holomorphic functions f:D→ℂ normalized by lvert f'(0)rvert=1 (equivalently, after rotation, f'(0)=1).
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
Degree at most d functions f:lbrace ± 1rbrace^n→ℝ have Fourier–Walsh expansion f(x)=Σ_S⊆ [n], |S|≤ d widehat f(S) x^S, x^S:=Π_i∈ Sx_i, [n]:=lbrace 1,…,nrbrace. For d∈ℕ set p_d:=2d/d+1.
Let D^d := z=(z_1,…,z_d)∈ℂ^d: lvert z_1rvert,…,lvert z_drvert<1 be the unit polydisc, and let the Schur class S_d be the set of analytic functions f:D^dtoD.
For any 1 ≤ p < ∞ and n ≥ 2, let C(p,n) be the smallest constant such that for any complex polynomial f of degree n with zeroes z_1,…,z_n satisfying 1/n Σ_i=1^n |z_i|^p ≤ 1, and every zero f(ζ)=0 of f, there exists a critical point f'(ξ) = 0 of f with |ξ - ζ| ≤ C(p,n). What is C(p,n)?
Let p_n denote the n-th prime. The bounded prime gap constant is C_88a = H_1 := liminf_n → ∞ (p_n+1 - p_n), the least limit point of the sequence of gaps between consecutive primes.
Let Ω⊂ℂ be simply connected with at least two boundary points in the extended complex plane, and let φ:ΩtoD be a conformal map. Brennan's conjecture states that ∫_Ωlvert φ'(z)rvert^p dx dy < ∞ qquadwhenever 4/3<p<4.
C_16 = L is the smallest constant for which the sharp Brezis–Gallouet inequality ‖u‖_L^∞(T^2)^2 ≤ 1/4π ‖∇ u‖_L^2(T^2)^2 Bigl[lnδ(u) + lnbigl(1+lnδ(u)bigr) + LBigr] holds for all zero-mean functions u ∈ H^2(T^2) with sufficiently large frequency ratio δ(u) := ‖Δ u‖_L^2(T^2)^2/‖∇ u‖_L^2(T^2)^2.
C_81a, Brun's Constant, is the sum of the reciprocals of the twin primes.
A Dirichlet character of level q is an arithmetic function χ that is multiplicative, is defined by a character on (ℤ/qℤ)^ast on integers coprime to q, and is 0 on integers not coprime to q.
In ℝ^d (d≥ 1), let M_d denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by M_d f(x) := sup_r>0 1/lvert Q(x,r)rvert∫_Q(x,r) lvert f(y)rvert dy, where Q(x,r) is a closed ℓ_∞ ball of radius r and center x in ℝ^d, that is, a closed cube centered at x, with sides…
Let λ_n,2 be the random variable assigning two uniformly random binary strings of length n the length of their longest common subsequence. Then C_31a is the (well-defined) limit C_31a := lim_n → inftyE[λ_n,2]/n.
C_8 = R is the least constant such that there are no zeroes σ+it of the Riemann zeta function with lvert t rvert ≥ 2 and σ > 1 - 1/R log lvert t rvert.
The constant term of one-shot channel simulation [HJMR07], [BG14], [LEG18], [Li25] is given as (we use the definition in [Li25]) C_32=limsup_t→∞(sup_p_X,Y: I(X;Y)=t inf_p_S|X,Y: I(X;S)=0H(Y|S)-t-log_2t), where H(Ylvert S)=H(Y,S)-H(S) is the conditional entropy (in bits), and I(X;Y)=H(X)+H(Y)-H(X,Y)…
In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.
In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈ℕ such that every sequence S over G of length lvert Srvert≥ l has a non-empty zero-sum subsequence.
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
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.
In algebraic complexity theory, for each real k ≥ 0, let ω(k) denote the exponent for multiplying an n × n^k matrix by an n^k × n matrix. We define α := supk ≥ 0 : ω(k) = 2.
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 f:\-1,1\^n→\-1,1\ be a Boolean function with Fourier expansion f(x)=Σ_S⊆[n]hat f(S)χ_S(x). Its spectral entropy is H(hat f^2) := Σ_S⊆[n]hat f(S)^2log_21/hat f(S)^2, and its total influence is Inf(f) := Σ_S⊆[n]hat f(S)^2 lvert Srvert.
