Vela Mathematics Program: problems

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1,217 problems · 7/25Number SourcesGraph view

declared openerdos:301

declared open · formalizederdos:302

∀ (f : ℕ → ℕ), (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) → Filter.Tendsto (fun N => ↑(f N) / ↑N) Filter.atTop (nhds sorry)

number theoryunit fractionsA3903953 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 open · formalizederdos:304

∀ {a b : ℕ}, 1 ∈ Erdos304.unitFractionExpressible a b → a ∣ b

number theoryunit fractionsA097847 · A0978492 sources

declared provederdos:305

Erdős problem 305

number theoryunit fractionspossible

declared open · formalizederdos:306

a/b \in \mathbb{Q}_{>0}

and

b

is squarefree, can

a/b

always be written as a finite sum of reciprocals of distinct products of two distinct primes?

number theoryunit fractionsN/A4 sources

declared verifiable · formalizederdos:307

sorry ↔ ∃ P Q, (∀ p ∈ P, Nat.Prime p) ∧ (∀ q ∈ Q, Nat.Prime q) ∧ 1 = (∑ p ∈ P, (↑p)⁻¹) * ∑ q ∈ Q, (↑q)⁻¹

number theoryunit fractionsN/A2 sources

declared provederdos:308

Erdős problem 308

number theoryunit fractionspossible

declared disprovederdos:309

Erdős problem 309

number theoryunit fractionsA217693

declared provederdos:310

Erdős problem 310

number theoryunit fractionsN/A

declared openerdos:311

Erdős problem 311

number theoryunit fractionsN/A

declared open · formalizederdos:312

True ↔ ∃ c, 0 < c ∧ ∀ (K : ℝ), 1 < K → ∃ N₀, ∀ (n : ℕ) (a : Fin n → ℕ), n ≥ N₀ ∧ ∑ i, (↑(a i))⁻¹ > K → ∃ S, 1 - Real.exp (-(c * K)) < ∑ i ∈ S, (↑(a i))⁻¹ ∧ ∑ i ∈ S, (↑(a i))⁻¹ ≤ 1

number theoryunit fractionsN/A2 sources

declared open · formalizederdos:313

sorry ↔ Erdos313.erdos313Solutions.Infinite

number theoryunit fractionsA0543772 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 disproved (Lean) · formalizederdos:316

False ↔ ∀ (A : Finset ℕ), 0 ∉ A → 1 ∉ A → ∑ n ∈ A, 1 / ↑n < 2 → ∃ A₁ A₂, Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧ ∑ n ∈ A₁, 1 / ↑n < 1 ∧ ∑ n ∈ A₂, 1 / ↑n < 1

number theoryunit fractionsN/A4 sources

declared open · formalizederdos:317

∀ᶠ (n : ℕ) in Filter.atTop, ∀ (δ : Fin n → ℚ), δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| ≥ 1 / ↑((Finset.Icc 1 n).lcm id)

number theoryunit fractionsN/A2 sources

declared solved · formalizederdos:318

∃ A, A.HasPosDensity ∧ ¬Erdos318.P₁ A

number theoryunit fractionsN/A2 sources

declared open · formalizederdos:319

∀ (N : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), A.card = x} sorry

number theoryunit fractionspossible2 sources

declared solvederdos:320

Let

S(N)

count the distinct values of

\sum_{n\in A} 1/n

over

A\subseteq\{1,\dots,N\}

. Estimate

S(N)

number theoryunit fractionsA0722072 sources

declared solved · formalizederdos:321

What is the largest

A\subseteq\{1,\dots,N\}

such that all subset sums

\sum_{n\in S}1/n

(over

S\subseteq A

) are distinct?

number theoryunit fractionsA384927 · A3915923 sources

declared openerdos:322

Erdős problem 322

number theorypowersA025456 · A025418

declared open · formalizederdos:323

True ↔ ∀ k ≥ 1, ∀ ε > 0, (fun x => ↑x ^ (1 - ε)) =O[Filter.atTop] fun x => ↑(Erdos323.f k k x)

number theorypowersA004825 · A004831 · A004832 · A004833 · A004842 · A004843 · A004844 · A004845 · A004857 · A004869 · possible2 sources

declared open · formalizederdos:324

sorry ↔ ∃ f, Set.InjOn (fun x => match x with | (a, b) => Polynomial.eval (↑a) f + Polynomial.eval (↑b) f) {(a, b) | a < b}

number theorypowersN/A2 sources

declared open · formalizederdos:325

True ↔ ∀ (k : ℕ), 3 ≤ k → (fun x => ↑x ^ (3 / ↑k)) =O[Filter.atTop] fun x => ↑(Erdos325.cardIsSumThreePowerBelow k x)

number theorypowersA004825 · A004832 · A004843 · A004854 · A004865 · possible2 sources

declared open · formalizederdos:326

Does there exist

A=\{a_1<a_2<\cdots\}\subset \mathbb{N}

which is a minimal basis of order

2

(every large integer is the sum of

2

elements from

A

, and no proper subset of

A

has this property) such that

\lim_{k\to \infty}a_k/k^2=c

for some

c\neq 0

? A claimed construction gives a minimal basis with

A(x)=C\sqrt{x}+O(1)

, answering the question affirmatively; Erdős and Graham had conjectured a negative answer.

number theoryadditive basisN/A4 sources

declared openerdos:327

Erdős problem 327

number theoryunit fractionsA3849272 sources

declared disproved (Lean) · formalizederdos:328

False ↔ ∀ (C : ℕ), 0 < C → ∃ t, ∀ (A : Set ℕ), (∀ (n : ℕ), AdditiveCombinatorics.sumRep A n ≤ C) → ∃ P, ⋃ i, P i = A ∧ Set.univ.PairwiseDisjoint P ∧ ∀ (i : Fin t) (n : ℕ), AdditiveCombinatorics.sumRep (P i) n < C

number theoryadditive combinatoricsN/A4 sources

declared open · formalizederdos:329

sSup {x | ∃ A, ∃ (_ : IsSidon A), Erdos329.sidonUpperDensity A = x} = sorry

number theorypossible2 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 disproved (Lean) · formalizederdos:331

False ↔ ∀ (A B : Set ℕ), ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count A n)) → ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count B n)) → {(a₁, a₂, b₁, b₂) | a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧ a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁}.Infinite

number theoryadditive combinatoricsN/A5 sources

declared open · formalizederdos:332

∀ (A : Set ℕ), sorry A → Erdos332.HasBoundedGaps (Erdos332.D_A A)

number theoryN/A3 sources

declared disproved (Lean) · formalizederdos:333

False ↔ ∀ (A : Set ℕ), A.HasDensity 0 → ∃ B, A ⊆ B + B ∧ (fun N => ↑(B ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => √↑N

number theoryadditive basisN/A6 sources

declared openerdos:334

Erdős problem 334

number theoryA062241 · A045535

declared openerdos:335

Erdős problem 335

number theoryadditive combinatoricsN/A2 sources

declared openerdos:336

h(r)

is the maximal finite exact order attainable by an additive basis of order at most

r

, what is

\lim_{r \to \infty} h(r)/r^2

? The candidate proof identifies the sharp limit

1/3

number theoryadditive basispossible2 sources

declared disproved (Lean) · formalizederdos:337

False ↔ ∀ (A : Set ℕ), A.IsAsymptoticAddBasis → ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) → Filter.Tendsto (fun N => ↑((A + A) ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop Filter.atTop

number theoryadditive combinatoricsadditive basisN/A5 sources

declared openerdos:338

Erdős problem 338

number theoryadditive basisN/A

declared provederdos:339

Erdős problem 339

number theoryadditive basisN/A

declared open · formalizederdos:340

∀ ε > 0, (fun n => √↑n / ↑n ^ ε) =O[Filter.atTop] fun n => ↑(Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard

number theoryadditive combinatoricssidon setsA080200 · A0052822 sources

declared open · formalizederdos:341

True ↔ ∀ (a : ℕ → ℤ), (∀ᶠ (n : ℕ) in Filter.atTop, IsLeast {x | a n < x ∧ x ∉ {x | ∃ i ≤ n, ∃ j ≤ n, a i + a j = x}} (a (n + 1))) → have d := fun i => a (i + 1) - a i; ∃ p > 0, ∀ᶠ (m : ℕ) in Filter.atTop, d (m + p) = d m

number theoryN/A2 sources

declared open · formalizederdos:342

sorry ↔ ∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → {n | ∃ m, a m = a n + 2}.Infinite

number theoryA0028583 sources

declared provederdos:343

Erdős problem 343

number theorycomplete sequencesN/A

declared provederdos:344

Erdős problem 344

number theorycomplete sequencesN/A

declared openerdos:345

Erdős problem 345

number theorycomplete sequencesA0016612 sources

declared open · formalizederdos:346

Let

A=\{1\leq a_1< a_2<\cdots\}

be a set of integers such that

A\backslash B

is complete for any finite subset

B

and not complete for any infinite subset

B

. If

a_{n+1}/a_n \geq 1+\epsilon

for all

n

, must

\lim_n a_{n+1}/a_n=(1+\sqrt{5})/2

? Under the reading where the ratio limit is assumed to exist, a Lean-verified argument forces the limit to be the golden ratio; a separate construction disproves the literal statement where convergence is not assumed.

number theorycomplete sequencesN/A4 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 open · formalizederdos:348

{x | ∃ m n, ∃ (_ : m < n), ∃ a, ∃ (_ : Monotone a) (_ : ∀ (s : Finset ℕ), s.card = m → IsAddComplete (Set.range (Function.updateFinset a s 0))) (_ : ∀ (t : Finset ℕ), t.card = n → ¬IsAddComplete (Set.range (Function.updateFinset a t 0))), (m, n) = x} = sorry

number theorycomplete sequencesN/A2 sources

declared open · formalizederdos:349

∀ (t α : ℝ), 0 < t → 2 < α → ¬Erdos349.IsGoodPair t α

number theorycomplete sequencesN/A3 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

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