Vela Mathematics Program: problems

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111 problems · 1/3Number SourcesGraph view

declared proved (Lean) · formalizederdos:24

True ↔ ∀ (n : ℕ) (G : SimpleGraph (Fin (5 * n))), G.CliqueFree 3 → G.copyCount (SimpleGraph.cycleGraph 5) ≤ n ^ 5

graph theorypossible5 sources

declared proved (Lean) · formalizederdos:31

∀ (A : Set ℕ), A.Infinite → ∃ B, B.HasDensity 0 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, n ∈ A + B

number theoryadditive basisN/A5 sources

declared proved (Lean) · formalizederdos:38

True ↔ ∃ B, ¬B.IsWeakAddBasis ∧ ∃ f, (∀ (α : ℝ), 0 < α → α < 1 → f α > 0) ∧ ∀ (A : Set ℕ) (N : ℕ), have α := schnirelmannDensity A; ∃ b ∈ B, ↑(Set.Ioc 0 N ∩ (A ∪ (A + {b}))).ncard ≥ (α + f α) * ↑N

number theoryN/A6 sources

declared proved (Lean) · formalizederdos:45

True ↔ ∀ (k : ℕ), 2 ≤ k → ∃ n, ∀ (colouring : ℕ → Fin k), ∃ colour, ∃ D' ⊆ {d ∈ n.divisors | 1 < d ∧ d < n}, (∀ d ∈ D', colouring d = colour) ∧ D'.reciprocalSum = 1

number theoryunit fractionsramsey theorypossible4 sources

declared proved (Lean) · formalizederdos:46

True ↔ ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite → ∃ S, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = 1 ∧ (𝓒 '' ↑S).Subsingleton

number theoryunit fractionsramsey theoryN/A4 sources

declared proved (Lean) · formalized · $100erdos:47

True ↔ ∀ (δ : ℝ), 0 < δ → ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, δ * Real.log ↑N < A.reciprocalSum → ∃ S ⊆ A, S.reciprocalSum = 1

number theoryunit fractionsN/A4 sources

declared proved (Lean) · formalizederdos:71

True ↔ ∀ (P : Set ℕ), P.IsAPOfLength ⊤ → (∃ n ∈ P, Even n) → ∃ c, ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj], c ≤ G.averageDegree → ∃ v w, w.IsCycle ∧ w.length ∈ P

graph theorycyclesN/A4 sources

declared proved (Lean) · formalizederdos:93

∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.ConvexIndep ↑A → A.card / 2 ≤ EuclideanGeometry.distinctDistances A

geometryconvexdistancesN/A5 sources

declared proved (Lean) · formalized · £25erdos:94

∃ C > 0, ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.ConvexIndep ↑P → ∑ u ∈ EuclideanGeometry.distanceSet P, ↑(EuclideanGeometry.distanceMultiplicity P u) ^ 2 ≤ C * ↑P.card ^ 3

geometryconvexdistancesA3878585 sources

declared proved (Lean) · formalizederdos:115

True ↔ ∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (p : Polynomial ℂ), p.Monic → p.natDegree = n → IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} → ∀ (z : ℂ), ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2

polynomialsanalysisN/A5 sources

declared proved (Lean) · formalized · $250erdos:123

Let

a,b,c>1

be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form

a^k b^l c^m

(

k,l,m\ge 0

), none dividing another?

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:134

True ↔ ∀ (ε δ : ℝ), 0 < ε → 0 < δ → ∃ N, ∀ n ≥ N, ∀ (G : SimpleGraph (Fin n)), G.CliqueFree 3 → (∀ (v : Fin n), ↑(G.degree v) < (↑n).rpow (1 / 2 - ε)) → ∃ H, G ≤ H ∧ H.CliqueFree 3 ∧ (∀ (x y : Fin n), x ≠ y → H.Adj x y ∨ ∃ z, H.Adj x z ∧ H.Adj z y) ∧ ↑(H.edgeFinset \ G.edgeFinset).card ≤ δ * ↑n ^ 2

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:150

True ↔ ∃ α < 2, Filter.Tendsto (fun n => ↑(Erdos150.maxMinimalCuts n) ^ (1 / ↑n)) Filter.atTop (nhds α)

graph theorypossible5 sources

declared proved (Lean) · formalizederdos:154

True ↔ ∀ (m : ℕ), 2 ≤ m → ∀ (N : ℕ → ℕ) (A : ℕ → Finset ℕ), Filter.Tendsto (fun k => ↑(N k)) Filter.atTop Filter.atTop → (∀ (k x : ℕ), x ∈ A k → x ≤ N k) → (∀ (k : ℕ), IsSidon ↑(A k)) → Filter.Tendsto (fun k => ↑(A k).card / √↑(N k)) Filter.atTop (nhds 1) → ∀ i < m, Filter.Tendsto (fun k => ↑{s ∈ A k + A k | s % m = i}.card / ↑(A k + A k).card) Filter.atTop (nhds (1 / ↑m))

sidon setsN/A6 sources

declared proved (Lean) · formalizederdos:164

True ↔ ∀ (A : Set ℕ), (∀ a ∈ A, 2 ≤ a) → Erdos1196.IsPrimitive A → ∑' (a : ↑A), 1 / (↑↑a * Real.log ↑↑a) ≤ ∑' (p : ↑{p | Nat.Prime p}), 1 / (↑↑p * Real.log ↑↑p)

number theoryprimitive setsA1372455 sources

declared proved (Lean) · formalizederdos:178

True ↔ ∀ (a : ℕ → ℕ → ℕ), (∀ (i : ℕ), StrictMono (a i)) → ∃ f, (∀ (n : ℕ), f n = 1 ∨ f n = -1) ∧ ∀ (d : ℕ), ∃ C, ∀ (m i : ℕ), i < d → |∑ j ∈ Finset.range m, f (a i j)| ≤ ↑C

discrepancyN/A5 sources

declared proved (Lean) · formalizederdos:214

True ↔ ∀ (S : Set (EuclideanSpace ℝ (Fin 2))), Erdos214.UnitDistanceAvoiding S → ∃ p, (∀ (i : Fin 4), p i ∈ Sᶜ) ∧ Congruent p Erdos214.unitSquare

geometrydistancesN/A5 sources

declared proved (Lean) · formalizederdos:221

True ↔ ∃ A, ((fun N => ↑{a | a ∈ A ∧ a ≤ N}.ncard) =O[Filter.atTop] fun N => ↑N / Real.log ↑N) ∧ ∀ᶠ (N : ℕ) in Filter.atTop, ∃ k a, 0 ≤ k ∧ a ∈ A ∧ N = 2 ^ k + a

number theoryadditive basisN/A5 sources

declared proved (Lean) · formalizederdos:224

∀ {d : ℕ} (A : Finset (EuclideanSpace ℝ (Fin d))), A.card = 2 ^ d + 1 → ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ Erdos224.ObtuseAt x y z

geometryN/A5 sources

declared proved (Lean) · formalizederdos:226

True ↔ ∃ F, Differentiable ℂ F ∧ (∀ (x : ℝ), (F ↑x).im = 0) ∧ (∀ (g : ℝ →ᵃ[ℝ] ℝ), (fun x => (F ↑x).re) ≠ ⇑g) ∧ Erdos226.PreservesRationality fun x => (F ↑x).re

analysisN/A5 sources

declared proved (Lean) · formalizederdos:229

True ↔ ∀ (S : ℕ → Set ℂ), (∀ (n : ℕ), derivedSet (S n) = ∅) → ∃ f, Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1, ∃ k, ∀ z ∈ S n, iteratedDeriv k f z = 0

analysisiterated functionsN/A5 sources

declared proved (Lean) · formalizederdos:246

∀ (a b : ℕ), 2 ≤ a → 2 ≤ b → a.Coprime b → IsAddComplete (Erdos246.Gamma a b)

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:258

True ↔ ∀ (a : ℕ → ℕ), (∀ (n : ℕ), 2 ≤ a n) → Filter.Tendsto a Filter.atTop Filter.atTop → Irrational (∑' (n : ℕ), ↑(n + 1).divisors.card / ↑(∏ i ∈ Finset.Icc 1 (n + 1), a i))

irrationalityN/A6 sources

declared proved (Lean) · formalizederdos:259

Irrational (∑' (n : ℕ), ↑(ArithmeticFunction.moebius n) ^ 2 * ↑n / 2 ^ n)

irrationalityA3711345 sources

declared proved (Lean) · formalizederdos:268

∀ (d : ℕ), (interior {x | ∃ A, A.Infinite ∧ (Summable fun n => 1 / ↑↑n) ∧ x = fun i => ∑' (n : ↑A), 1 / (↑↑n + ↑↑i)}).Nonempty

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:275

∀ (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ), (∃ k, ∀ x ∈ Set.Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD ↑(n i)]) → ∀ (x : ℤ), ∃ i, x ≡ a i [ZMOD ↑(n i)]

number theorycovering systemsN/A5 sources

declared proved (Lean) · formalizederdos:281

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → (∀ (i : ℕ), 0 < n i) → (∀ (a : Erdos281.ResidueChoice n), (Erdos281.avoidAll n a).HasIntDensity 0) → ∀ (ε : ℝ), 0 < ε → ∃ k, ∀ (a : Erdos281.ResidueChoice n), ∃ d, (Erdos281.avoidPrefix n a k).HasIntDensity d ∧ d < ε

number theorycovering systemsN/A7 sources

declared proved (Lean) · formalizederdos:283

True ↔ ∀ (p : Polynomial ℤ), Erdos283.Condition p

number theoryunit fractionsA3807916 sources

declared proved (Lean) · formalizederdos:290

True ↔ ∀ (a : ℕ), 1 ≤ a → ∃ b, a < b ∧ Erdos290.harmonicDen a (b + 1) < Erdos290.harmonicDen a b

number theoryunit fractionsA3750815 sources

declared proved (Lean) · formalizederdos:296

(∀ (N k : ℕ), Erdos296.HasDisjointUnitDecomps N k → ↑k ≤ Erdos296.recipSum (Finset.Icc 1 N)) ∧ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∀ᶠ (N : ℕ) in Filter.atTop, Erdos296.HasDisjointUnitDecomps N ⌊(1 - ε) * Real.log ↑N⌋₊

number theoryunit fractionspossible5 sources

declared proved (Lean) · formalizederdos:298

True ↔ ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1

number theoryunit fractionsN/A4 sources

declared proved (Lean) · formalizederdos:303

True ↔ ∀ (𝓒 : ℤ → ℤ), (Set.range 𝓒).Finite → ∃ a b c, [a, b, c, 0].Nodup ∧ 1 / ↑a = 1 / ↑b + 1 / ↑c ∧ (𝓒 '' {a, b, c}).Subsingleton

number theoryunit fractionsN/A5 sources

declared proved (Lean) · formalizederdos:314

True ↔ Filter.liminf (fun n => ↑n ^ 2 * Erdos314.epsilon n) Filter.atTop = 0

number theoryunit fractionsN/A5 sources

declared proved (Lean) · formalizederdos:315

True ↔ ∀ (a : ℕ → ℕ), (∀ (i : ℕ), 0 < a i) → StrictMono a → (∃ i, a i ≠ Erdos315.u i + 1) → ∑' (i : ℕ), 1 / ↑(a i) = 1 → Filter.liminf (fun i => ↑(a i) ^ (1 / 2) ^ (i + 1)) Filter.atTop < Erdos315.c₀

number theoryunit fractionsA000058 · A0763935 sources

declared proved (Lean) · formalizederdos:330

True ↔ ∃ A h, Erdos330.MinAsymptoticAddBasisOfOrder A h ∧ 0 < A.upperDensity ∧ ∀ n ∈ A, 0 < (Erdos330.UnrepWithout A n h).upperDensity

number theoryadditive basisN/A5 sources

declared proved (Lean) · formalizederdos:347

True ↔ ∃ a, Monotone a ∧ Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) Filter.atTop (nhds 2) ∧ ∀ (ι : ℕ → ℕ), (Set.range ι)ᶜ.Finite → (subsetSums (Set.range (a ∘ ι))).HasDensity 1

number theorycomplete sequencesN/A6 sources

declared proved (Lean) · formalizederdos:350

∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : DecidableEq M] (A : Finset M), Erdos350.DistinctSubsetSums ↑A ↔ Erdos350.DecidableDistinctSubsetSums A

number theoryadditive combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:351

True ↔ ∀ (P : Polynomial ℚ), 0 < P.natDegree → 0 < P.leadingCoeff → Erdos351.HasCompleteImage P

number theorycomplete sequencesN/A6 sources

declared proved (Lean) · formalizederdos:353

True ↔ ∀ (A : Set (EuclideanSpace ℝ (Fin 2))), MeasurableSet A → MeasureTheory.volume A = ⊤ → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, ∃ d ∈ A, EuclideanGeometry.IsIsoscelesTrapezoid a b c d ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c, d}) = 1

geometryN/A4 sources

declared proved (Lean) · formalizederdos:355

True ↔ ∃ A, IsLacunary A ∧ ∃ u v, u < v ∧ ∀ (q : ℚ), ↑q ∈ Set.Ioo u v → q ∈ {x | ∃ A', ∃ (_ : ↑A' ⊆ Set.range A), ∑ a ∈ A', 1 / ↑a = x}

number theoryunit fractionsN/A5 sources

declared proved (Lean) · formalizederdos:369

True ↔ ∀ (ε : ℝ), 0 < ε → ∀ (k : ℕ), 2 ≤ k → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧ ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, ↑p ≤ ↑n ^ ε

number theoryN/A7 sources

declared proved (Lean) · formalizederdos:370

True ↔ {n | ↑n.maxPrimeFac < √↑n ∧ ↑(n + 1).maxPrimeFac < √(↑n + 1)}.Infinite

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:379

Filter.limsup (fun n => ↑(Erdos379.S n)) Filter.atTop = ⊤

number theorybinomial coefficientspossible5 sources

declared proved (Lean) · formalizederdos:392

∀ (A : ℕ → ℕ), (∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) → (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n

number theoryfactorialspossible5 sources

declared proved (Lean) · formalizederdos:401

True ↔ ∃ f, Filter.Tendsto f Filter.atTop Filter.atTop ∧ ∀ (r : ℕ), 1 ≤ r → {n | ∃ a₁ a₂, 0 < a₁ ∧ 0 < a₂ ∧ ↑a₁ + ↑a₂ > ↑n + f r * Real.log ↑n ∧ a₁.factorial * a₂.factorial ∣ n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite

number theoryfactorialsN/A6 sources

declared proved (Lean) · formalizederdos:403

True ↔ {p | (∀ a ∈ p.2, 0 < a) ∧ 2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite

number theoryfactorialsN/A4 sources

declared proved (Lean) · formalizederdos:418

True ↔ {x | ∃ n, n - n.totient = x}ᶜ.Infinite

number theoryA005278 · A2639585 sources

declared proved (Lean) · formalizederdos:427

True ↔ Erdos427.erdos427

number theoryprimesN/A5 sources

declared proved (Lean) · formalizederdos:434

∀ (n k : ℕ), 1 ≤ n → 2 ≤ k → k ≤ n → IsGreatest {x | ∃ S, ∃ (_ : S ⊆ Finset.Icc 1 n) (_ : S.card = k) (_ : S.gcd id = 1), Erdos434.Nat.NcardUnrepresentable ↑S = x} (Erdos434.Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))

number theorypossible5 sources

declared proved (Lean) · formalizederdos:435

∀ (n : ℕ), n ≠ 0 → (∀ (p k : ℕ), Nat.Prime p → n ≠ p ^ k) → IsGreatest {m | ¬∃ c, m = ∑ i ∈ Finset.Ico 1 n, ↑(c i) * ↑(n.choose i)} (∑ p ∈ n.primeFactors, (∑ d ∈ Finset.Icc 1 (n.factorization p), ↑(n.choose (p ^ d))) * (↑p - 1) - ↑n)

number theoryA3894795 sources

Problems · Vela Mathematics Program

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