Vela Mathematics Program: problems

Problem ledger

1,217 problems · 14/25Number SourcesGraph view

declared disprovederdos:651

Erdős problem 651

geometryconvexpossible

declared provederdos:652

Erdős problem 652

geometrydistancesN/A2 sources

declared open · formalizederdos:653

True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)

geometrydistancespossible2 sources

declared openerdos:654

n

planar points have no four concyclic, must some point determine

(1 - o(1))n

distinct distances? Failing that, can one always force more than

(1/3 + c)n

geometrydistancespossible3 sources

declared open · formalizederdos:655

False ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (X : Finset (EuclideanSpace ℝ (Fin 2))), X.card = n → Erdos655.IsValid X → (1 + c) * ↑n / 2 ≤ ↑(EuclideanGeometry.distinctDistances X)

geometrydistancespossible3 sources

declared provederdos:656

Erdős problem 656

number theoryadditive combinatoricsN/A

declared openerdos:657

Erdős problem 657

geometrydistancespossible

declared proved (Lean)erdos:658

Erdős problem 658

additive combinatoricspossible4 sources

declared proved (Lean) · formalizederdos:659

True ↔ ∃ A, (∀ (n : ℕ), (A n).card = n ∧ ∀ S ⊆ A n, S.card = 4 → 3 ≤ EuclideanGeometry.distinctDistances S) ∧ (fun n => ↑(EuclideanGeometry.distinctDistances (A n))) =O[Filter.atTop] fun n => ↑n / √(Real.log ↑n)

geometrydistancespossible6 sources

declared openerdos:660

Erdős problem 660

geometrydistancesconvexpossible

declared open · $50erdos:661

Erdős problem 661

geometrydistancespossible

declared openerdos:662

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.

geometrydistancesN/A3 sources

declared openerdos:663

Erdős problem 663

number theoryA391668

declared disprovederdos:664

Erdős problem 664

combinatoricsN/A

declared openerdos:665

Erdős problem 665

combinatoricsN/A2 sources

declared disproved (Lean) · formalizederdos:666

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ∀ H ≤ SimpleGraph.hypercube n, ε * ↑n * 2 ^ (n - 1) ≤ ↑H.edgeSet.ncard → (SimpleGraph.cycleGraph 6).IsContained H

graph theorypossible5 sources

declared openerdos:667

Erdős problem 667

graph theoryramsey theoryN/A

declared openerdos:668

Erdős problem 668

geometrydistancesA385657

declared openerdos:669

Erdős problem 669

geometryA003035 · A006065 · A008997 · possible

declared openerdos:670

Erdős asked whether every

n

-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least

(1+o(1))n^2

. Disproved: an explicit high-dimensional construction beats the conjectured constant.

geometrydistancesN/A2 sources

declared open · $250erdos:671

For triangular arrays of nodes

a_i^n\in[-1,1]

let

\mathcal{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

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

and yet

\mathcal{L}^nf(x) \to f(x)

? Is there a choice with

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

for every

x

, yet for every continuous

f

some

x

has

\mathcal{L}^nf(x)\to f(x)

? Both questions are claimed resolved in the affirmative.

analysisN/A3 sources

declared verifiable · formalizederdos:672

True ↔ ∀ (k l : ℕ), l > 1 → k ≥ 4 → Erdos672.Erdos672With k l

number theoryN/A2 sources

declared provederdos:673

Erdős problem 673

number theorydivisorsN/A

declared proved (Lean) · formalizederdos:674

True ↔ Erdos674.solutionSet.Nonempty

number theoryN/A4 sources

declared openerdos:675

Erdős problem 675

number theoryN/A2 sources

declared openerdos:676

Erdős problem 676

number theoryA390181 · in progress

declared open · formalizederdos:677

∀ (m n k : ℕ), k > 0 → m ≥ n + k → Finset.lcmInterval m k ≠ Finset.lcmInterval n k

number theorypossible2 sources

declared proved (Lean) · formalizederdos:678

True ↔ ∀ᶠ (k : ℕ) in Filter.atTop, {(m, n) | n + k ≤ m ∧ Finset.lcmInterval m (k + 1) < Finset.lcmInterval n k}.Nonempty

number theorypossible5 sources

declared openerdos:679

Erdős problem 679

number theoryN/A2 sources

declared open · formalizederdos:680

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k, k ≠ 0 ∧ (n + k).minFac > k ^ 2 + 1

number theoryprimesN/A2 sources

declared open · formalizederdos:681

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k > 0, (n + k).Composite ∧ ∀ (p : ℕ), Erdos681.IsLPF p (n + k) → p > k ^ 2

number theoryprimesA3896802 sources

declared provederdos:682

Erdős problem 682

number theoryprimesA386978

declared open · formalizederdos:683

True ↔ ∃ c > 0, ∀ (n k : ℕ), 0 < k ∧ k < n → ↑(Erdos683.P n k) > min (↑n - ↑k + 1) (↑k ^ (1 + c))

number theoryprimesbinomial coefficientsA006530 · A074399 · A121359 · possible2 sources

declared openerdos:684

For the least

k

at which the small-prime part of

\binom{n}{k}

exceeds

n^2

, how large can

f(n)

be?

number theoryprimesbinomial coefficientsA392019 · possible3 sources

declared openerdos:685

Erdős problem 685

number theoryprimesbinomial coefficientsN/A

declared open · formalizederdos:686

sorry ↔ ∀ N ≥ 2, ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))

number theoryN/A4 sources

declared open · $1000erdos:687

Erdős problem 687

number theoryA048670 · A058989

declared open · formalizederdos:688

sorry =O[Filter.atTop] Erdos688.epsilonFunction

number theoryN/A2 sources

declared open · formalizederdos:689

sorry ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, ∀ m ∈ Finset.Icc 1 n, 2 ≤ {p ∈ Finset.Icc 1 n | Nat.Prime p ∧ a p ≡ m [MOD p]}.card

number theoryN/A3 sources

declared solvederdos:690

Erdős problem 690

number theorypossible4 sources

declared openerdos:691

Erdős problem 691

number theoryN/A

declared disproved (Lean) · formalizederdos:692

False ↔ ∀ (δ : ℕ → ℕ → ℝ), (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) → ∀ (n : ℕ), UnimodularOn (δ n) (n + 1)

number theorydivisorsN/A5 sources

declared openerdos:693

Erdős problem 693

number theorydivisorsA391118 · possible

declared solved (Lean) · formalizederdos:694

∀ (fmax fmin : ℕ → ℕ), (∀ (n : ℕ), IsGreatest (Nat.totient ⁻¹' {n}) (fmax n)) → (∀ (n : ℕ), IsLeast (Nat.totient ⁻¹' {n}) (fmin n)) → ∃ o, Filter.Tendsto o Filter.atTop (nhds 0) ∧ ∀ (x : ℕ), sSup {x_1 | ∃ n, ∃ (_ : n ≤ x) (_ : ∃ m, m.totient = n), ↑(fmax n) / ↑(fmin n) = x_1} = (Real.exp Real.eulerMascheroniConstant + o x) * Real.log (Real.log ↑x)

number theorypossible6 sources

declared open · formalizederdos:695

sorry ↔ ∀ {q : ℕ → ℕ}, StrictMono q → (∀ (i : ℕ), Nat.Prime (q i)) → (∀ (i : ℕ), q (i + 1) % q i = 1) → Filter.Tendsto (fun k => ↑(q k) ^ (1 / ↑k)) Filter.atTop Filter.atTop

number theoryA0610922 sources

declared solved (Lean)erdos:696

Erdős problem 696

number theorydivisorspossible4 sources

declared proved · formalizederdos:697

∀ (m : ℕ) (α : ℝ), ∃ δ, {n | ∃ d, d ≡ 1 [MOD m] ∧ ↑d ∈ Set.Ioo 1 (Real.exp (↑m ^ α)) ∧ d ∣ n}.HasDensity δ

number theorydivisorsN/A2 sources

declared proved (Lean) · formalizederdos:698

True ↔ ∃ h, Filter.Tendsto h Filter.atTop Filter.atTop ∧ ∀ (n i j : ℕ), 2 ≤ i → i < j → j ≤ n / 2 → h n ≤ (n.choose i).gcd (n.choose j)

number theorybinomial coefficientspossible5 sources

declared falsifiable · formalizederdos:699

True ↔ ∀ (n i j : ℕ), 1 ≤ i → i < j → j ≤ n / 2 → ∃ p, Nat.Prime p ∧ i ≤ p ∧ p ∣ (n.choose i).gcd (n.choose j)

number theorybinomial coefficientsN/A2 sources

declared open · formalizederdos:700

∀ (n : ℕ), ¬Nat.Prime n → 1 < n → (Erdos700.f n = n / Erdos700.P n ↔ True)

number theorybinomial coefficientsA091963 · possible2 sources

Problems · Vela Mathematics Program

Problem ledger

Search problems.science