Vela Mathematics Program: problems

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604 problems · 11/13Number SourcesGraph view

declared proved (Lean) · formalizederdos:974

∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ), z 0 = 1 → {k | ∀ j < n - 1, ∑ i, z i ^ (k + j) = 0}.Infinite → Erdos974.IsTuranConfiguration z

analysisN/A5 sources

declared open · formalizederdos:975

True ↔ ∀ (f : Polynomial ℤ), f.natDegree ≠ 0 → Irreducible f → (∀ᶠ (n : ℤ) in Filter.atTop, 1 ≤ Polynomial.eval n f) → ∃ c > 0, Filter.Tendsto (fun x => Erdos975.Erdos975Sum f x / (x * Real.log x)) Filter.atTop (nhds c)

number theorydivisorsA147807 · possible2 sources

declared open · formalizederdos:978

∀ {f : Polynomial ℤ}, Irreducible f → 2 < f.natDegree → (∀ (x : ℕ), f.natDegree ≠ 2 ^ x) → 0 < f.leadingCoeff → (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) → {n | Powerfree (f.natDegree - 1) (Polynomial.eval (↑n) f)}.HasPosDensity

number theorypossible2 sources

declared open · formalizederdos:979

True ↔ ∀ k ≥ 2, Filter.limsup (fun n => (Erdos979.solutionSet n k).encard) Filter.atTop = ⊤

number theoryA385316 · possible2 sources

declared falsifiable · formalizederdos:982

∀ (n : ℕ), 3 ≤ n → ∀ (p : Fin n → EuclideanSpace ℝ (Fin 2)), Function.Injective p → EuclideanGeometry.IsConvexPolygon p → ∃ i, {d | ∃ j, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2

geometryconvexdistancesA0045262 sources

declared open · formalizederdos:985

sorry ↔ ∀ (p : ℕ), Nat.Prime p → p ≠ 2 → ∃ q, Nat.Prime q ∧ q < p ∧ orderOf ↑q = p - 1

number theoryA002233 · A219429 · A103309 · possible2 sources

declared proved · formalizederdos:987

True ↔ ∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → Filter.limsup (fun k => Erdos987.A x k) Filter.atTop = ⊤

analysisdiscrepancyN/A4 sources

declared disproved (Lean) · formalizederdos:990

False ↔ ∃ C, ∀ (f : Polynomial ℂ), f.coeff 0 ≠ 0 → ∀ (α β : ℝ), 0 ≤ α → α ≤ β → β ≤ 2 * Real.pi → |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤ C * √(↑f.support.card * Real.log (Erdos990.M f))

analysisN/A6 sources

declared open · formalizederdos:996

Let

n_1<n_2<\cdots

be a lacunary sequence of integers and

f\in L^2([0,1])

with

n

th Fourier partial sum

f_n

. Is there an absolute constant

C>0

such that if

\| f-f_n\|_2 \ll (\log\log\log n)^{-C}

then

\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f

for almost every

\alpha

? A preprint answers this negatively via a dyadic spike-block counterexample.

analysisN/A4 sources

declared proved (Lean) · formalizederdos:997

True ↔ ∀ (α : ℝ), ¬Erdos997.IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))

analysisdiscrepancyprimesN/A6 sources

declared proved (Lean) · formalizederdos:1000

True ↔ ∃ n, StrictMono n ∧ 0 < n 0 ∧ Filter.Tendsto (Erdos1000.phiAvg n) Filter.atTop (nhds 0)

number theorydiophantine approximationN/A5 sources

declared open · formalizederdos:1002

sorry ↔ ∃ g, Monotone g ∧ Filter.Tendsto g Filter.atBot (nhds 0) ∧ Filter.Tendsto g Filter.atTop (nhds 1) ∧ ∀ (c : ℝ), Filter.Tendsto (fun n => (MeasureTheory.volume {α | α ∈ Set.Ioo 0 1 ∧ (fun α n => 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (α * ↑k))) α n ≤ c}).toReal) Filter.atTop (nhds (g c))

analysisdiophantine approximationN/A2 sources

declared open · formalizederdos:1003

True ↔ {n | n.totient = (n + 1).totient}.Infinite

number theoryA0012742 sources

declared open · formalizederdos:1004

True ↔ ∀ c > 0, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ n ≤ x, Erdos1004.IsDistinctTotientRun n ⌊Real.log ↑x ^ c⌋₊

number theorypossible3 sources

declared solved (Lean) · formalizederdos:1007

IsLeast {m | ∃ n G, G.HasDimension 4 ∧ G.edgeSet.ncard = m} sorry

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:1008

True ↔ ∃ c > 0, ∀ (V : Type) [Fintype V] (G : SimpleGraph V), ∃ H ≤ G, (SimpleGraph.cycleGraph 4).Free H ∧ c * ↑G.edgeSet.ncard ^ (2 / 3) ≤ ↑H.edgeSet.ncard

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:1014

∀ (k : ℕ), 3 ≤ k → Filter.Tendsto (fun l => ↑R(k, l + 1) / ↑R(k, l)) Filter.atTop (nhds 1)

graph theorypossible7 sources

declared proved (Lean) · formalizederdos:1022

False ↔ ∃ c, Filter.Tendsto c Filter.atTop Filter.atTop ∧ ∀ (t : ℕ), Erdos1022.SparseImpliesPropertyB t (c t)

combinatoricshypergraphspossible6 sources

declared solved (Lean) · formalizederdos:1023

True ↔ ∃ c, 0 < c ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos1023.F n)) fun n => c * 2 ^ n / ↑n ^ (1 / 2)

combinatoricsA347025 · possible5 sources

declared solved (Lean) · formalizederdos:1026

For a sequence of

n

distinct reals, determine the largest constant

c

such that some monotonic subsequence always has sum exceeding

(c-o(1))\cdot(1/\sqrt{n})

times the total sum. Resolved as

c = 1

combinatoricsA391431 · A3914906 sources

declared solved (Lean) · formalizederdos:1028

(fun n => ↑(Erdos1028.H n)) =Θ[Filter.atTop] fun n => ↑n ^ (3 / 2)

graph theorydiscrepancypossible5 sources

declared disproved (Lean) · formalizederdos:1034

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), ↑n ^ 2 / 4 < ↑G.edgeSet.ncard → ∃ T, G.IsNClique 3 T ∧ ∃ Y, Erdos1034.JoinedToTwo G T Y ∧ (1 / 2 - ε) * ↑n < ↑Y.card

graph theorypossible5 sources

declared proved (Lean) · formalizederdos:1036

True ↔ ∀ (c : ℝ), 0 < c → ∃ δ, 0 < δ ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), ↑G.cliqueNum ≤ c * Real.log ↑n → ↑G.indepNum ≤ c * Real.log ↑n → Erdos1036.HasManyNonIsomorphicInducedSubgraphs G (2 ^ (δ * ↑n))

graph theorypossible5 sources

declared disproved (Lean) · formalizederdos:1037

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ (C : ℝ), ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), (∀ (d : ℕ), {v | (G.neighborSet v).ncard = d}.ncard ≤ 2) → (1 / 2 + ε) * ↑n < ↑(Set.range fun v => (G.neighborSet v).ncard).ncard → ∃ s, Erdos1037.IsTrivialSet G s ∧ C * Real.log ↑n < ↑s.ncard

graph theoryN/A5 sources

declared open · formalizederdos:1038

Among all nonconstant monic polynomials

f

whose roots lie in

[-1, 1]

, determine

\inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}|

analysisN/A4 sources

declared falsifiable · formalizederdos:1041

∀ (n : ℕ) (f : Polynomial ℂ), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ z₁ z₂, ∃ (_ : {z₁, z₂} ≤ f.roots), ∃ γ, Set.range ⇑γ ⊆ {z | ‖Polynomial.eval z f‖ < 1} ∧ Erdos1041.length (Set.range ⇑γ) < 2

analysisN/A3 sources

declared disproved (Lean) · formalizederdos:1043

False ↔ ∀ (f : Polynomial ℂ), f.Monic → f.degree ≥ 1 → ∃ u, ‖u‖ = 1 ∧ MeasureTheory.volume (⇑(ℝ ∙ u).orthogonalProjection '' Erdos1043.levelSet f) ≤ 2

analysisN/A5 sources

declared solved (Lean) · formalizederdos:1044

IsGLB {L | ∃ f, Erdos1044.IsAdmissible f ∧ Erdos1044.maxBoundaryLength f = L} 2

analysisN/A5 sources

declared disproved (Lean) · formalizederdos:1047

False ↔ ∀ (f : Polynomial ℂ) (m : ℕ) (c : ℝ), f.Monic → (f.rootSet ℂ).ncard = m → 0 < c → (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = m → ∀ t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c), Convex ℝ t

analysisN/A5 sources

declared disproved (Lean) · formalizederdos:1048

False ↔ ∀ (r : ℝ) (f : Polynomial ℂ), r < 2 → f.Monic → f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) → ∃ z ∈ Erdos1048.openLevelSet f, ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)

analysisN/A5 sources

declared open · formalizederdos:1049

True ↔ ∀ t > 1, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))

irrationalityN/A2 sources

declared proved (Lean) · formalizederdos:1051

True ↔ ∀ (a : ℕ → ℤ), StrictMono a → Erdos1051.GrowthCondition a → Irrational (Erdos1051.ErdosSeries a)

irrationalityN/A6 sources

declared open · formalized · $10erdos:1052

sorry ↔ {n | Erdos1052.IsUnitaryPerfect n}.Finite

number theoryA0028272 sources

declared open · formalizederdos:1054

sorry ↔ (fun n => ↑(Erdos1054.f n)) =o[Filter.atTop] fun n => ↑n

number theorydivisorsA1674853 sources

declared open · formalizederdos:1055

∀ (r : ℕ+), {p | Nat.Prime p ∧ Erdos1055.IsOfClass r p}.Infinite

number theoryprimesA0051132 sources

declared open · formalizederdos:1056

True ↔ ∀ k ≥ 2, ∃ p, ∃ (_ : Nat.Prime p), ∃ boundaries, ∃ (_ : StrictMono boundaries), Erdos1056.AllModProdEqualsOne p boundaries

number theoryA0604273 sources

declared open · formalizederdos:1057

True ↔ Filter.Tendsto (fun x => Real.log (Erdos1057.carmichaelCounting x) / Real.log x) Filter.atTop (nhds 1)

number theoryA0069312 sources

declared open · formalizederdos:1059

Erdos1059.AllFactorialSubtractionsComposite 101

number theoryprimesA0641522 sources

declared open · formalizederdos:1060

∃ h, (h =o[Filter.atTop] fun n => 1 / Real.log (Real.log ↑n)) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑{k ∈ Finset.Iic n | k * (ArithmeticFunction.sigma 1) k = n}.card ≤ ↑n ^ h n

number theoryA3271532 sources

declared open · formalizederdos:1061

Let

S(x)

count ordered pairs

(a,b)

with

a+b \le x

and

\sigma(a)+\sigma(b) = \sigma(a+b)

. Erdos asked whether

S(x) \sim cx

. The opposite extreme holds: for every

R > 0

S(x)/(x(\log x)^R) \to \infty

, so the count beats every fixed logarithmic scale.

number theoryA110177 · possible4 sources

declared open · formalizederdos:1062

(∃ l, Filter.Tendsto (fun n => ↑(Erdos1062.f n) / ↑n) Filter.atTop (nhds l) ∧ Irrational l) ↔ True

number theoryA0383723 sources

declared open · formalizederdos:1063

have upper_bound := sorry; (fun k => ↑(Erdos1063.n k)) =O[Filter.atTop] upper_bound ∧ upper_bound =o[Filter.atTop] fun k => ↑k * (Finset.Icc 1 (k - 1)).lcm fun n => ↑n

number theoryA3893603 sources

declared proved · formalizederdos:1064

{n | n.totient > (n - n.totient).totient}.HasDensity 1

number theoryA051488 · A0514872 sources

declared open · formalizederdos:1065

True ↔ {p | ∃ q k, Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * q + 1}.Infinite

number theoryA074781 · A3394652 sources

declared disproved (Lean) · formalizederdos:1067

False ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = Cardinal.aleph 1 → ∃ H, H.coe.chromaticCardinal = Cardinal.aleph 1 ∧ H.coe.InfinitelyConnected

graph theoryset theoryN/A5 sources

declared open · formalizederdos:1068

True ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = Cardinal.aleph 1 → ∃ s, s.Countable ∧ (SimpleGraph.induce s G).InfinitelyConnected

graph theoryset theorychromatic numberN/A2 sources

declared proved (Lean) · formalizederdos:1071

True ↔ ∃ S, Maximal (fun T => (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1.ofLp 0 ∈ Set.Icc 0 1 ∧ seg.1.ofLp 1 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 0 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 1 ∈ Set.Icc 0 1) ∧ (↑T).Pairwise Erdos1071.SegmentsDisjoint) S

geometryN/A5 sources

declared open · formalizederdos:1072

True ↔ {p | Nat.Prime p ∧ Erdos1072.f p = p - 1}.Infinite

number theoryA073944 · A072937 · A1545542 sources

declared open · formalizederdos:1073

True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ (x : ℕ), Erdos1073.A x ≤ ↑x ^ o x

number theoryA2565192 sources

declared open · formalizederdos:1074

sorry ↔ ∃ c, Erdos1074.EHSNumbers.HasDensity c

number theoryA063980 · A0641642 sources

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