Vela Mathematics Program: problems

Narrowed toopen ×source:erdos-problems ×Clear all

Problem ledger

604 problems · 3/13Number SourcesGraph view

declared openerdos:201

Erdős problem 201

additive combinatoricsarithmetic progressionsA003002 · A003003 · A003004 · A003005 · possible

declared open · formalizederdos:203

True ↔ ∃ m, m.Coprime 6 ∧ ∀ (k l : ℕ), ¬Nat.Prime (2 ^ k * 3 ^ l * m + 1)

primescovering systemsN/A2 sources

declared open · formalizederdos:208

True ↔ ∀ ε > 0, (fun n => ↑(Erdos208.erdos208.s (n + 1)) - ↑(Erdos208.erdos208.s n)) =O[Filter.atTop] fun n => ↑(Erdos208.erdos208.s n) ^ ε

number theoryA005117 · A0762592 sources

declared open · formalizederdos:212

True ↔ ∃ u, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast

geometrydistancesN/A2 sources

declared open · formalizederdos:213

True ↔ ∀ n ≥ 4, Erdos213.Erdos213For n

geometrydistancesN/A2 sources

declared openerdos:217

Erdős problem 217

geometrydistancespossible

declared open · formalizederdos:218

{n | primeGap (n + 1) ≤ primeGap n}.HasDensity (1 / 2)

number theoryprimesA333230 · A333231 · A0641133 sources

declared openerdos:222

Erdős problem 222

number theorysquaresA001481 · A256435

declared open · formalizederdos:233

(fun N => ∑ n ∈ Finset.range N, ↑(primeGap n) ^ 2) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ 2

number theoryprimesA0747413 sources

declared open · formalizederdos:234

True ↔ ∃ f, Continuous f ∧ ∀ (c : NNReal), {n | ↑(primeGap n) / Real.log ↑n < ↑c}.HasDensity (f c)

number theoryprimesN/A2 sources

declared open · formalizederdos:236

(fun n => ↑(Erdos236.f n)) =o[Filter.atTop] fun n => Real.log ↑n

number theoryprimesA039669 · A1099252 sources

declared open · formalizederdos:238

True ↔ ∀ c₁ > 0, ∀ c₂ > 0, ∀ᶠ (x : ℝ) in Filter.atTop, ∃ k, c₁ * Real.log x < ↑k ∧ ∃ f m, (∀ (i : Fin k), ↑(f i) ≤ x ∧ f i = Nat.nth Nat.Prime (m + ↑i)) ∧ ∀ (i : Fin (k - 1)), c₂ < primeGap (m + ↑i)

number theoryprimesN/A2 sources

declared open · formalized · $100erdos:241

True ↔ Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos241.f N 3)) fun N => ↑N ^ (1 / 3)

additive combinatoricssidon setsA3877042 sources

declared open · formalizederdos:243

∀ (a : ℕ → ℕ), StrictMono a → Filter.Tendsto (fun n => ↑(a n) / ↑(a (n - 1)) ^ 2) Filter.atTop (nhds 1) → (Summable fun x => 1 / ↑(a x)) → ∀ᶠ (n : ℕ) in Filter.atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1

number theoryA0000582 sources

declared open · formalizederdos:244

True ↔ ∀ C > 1, 0 < {x | ∃ p k, ∃ (_ : Nat.Prime p), p + ⌊C ^ k⌋₊ = x}.lowerDensity

number theoryprimesN/A2 sources

declared open · formalizederdos:247

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → Filter.limsup (fun k => ↑(n k) / ↑k.succ) Filter.atTop = ⊤ → Transcendental ℚ (∑' (k : ℕ), 1 / 2 ^ n k)

number theoryirrationalityN/A2 sources

declared open · formalizederdos:249

True ↔ Irrational (∑' (n : ℕ), ↑n.totient / 2 ^ n)

number theoryirrationalityA2569362 sources

declared open · formalizederdos:251

True ↔ Irrational (∑' (n : ℕ), ↑(Nat.nth Nat.Prime n) / 2 ^ n)

number theoryirrationalityA0989903 sources

declared open · formalizederdos:252

True ↔ ∀ k ≥ 1, Irrational (Erdos252.erdos_252_sum k)

number theoryirrationalityA227988 · A227989 · A307036 · A359060 · possible2 sources

declared open · formalizederdos:254

A \subseteq \mathbb{N}

has unbounded dyadic-shell counts and

\sum_{n \in A} \|\theta n\| = \infty

for every

0 < \theta < 1

, must

A

be complete - is every sufficiently large integer a sum of distinct elements of

A

number theoryN/A4 sources

declared openerdos:256

Erdős problem 256

analysisN/A

declared open · formalizederdos:257

True ↔ ∀ (A : Set ℕ), A.Infinite → Irrational (∑' (n : ↑A), 1 / (2 ^ ↑n - 1))

irrationalityN/A2 sources

declared open · formalizederdos:260

sorry ↔ ∀ (a : ℕ → ℤ) (s : ℝ), StrictMono a → Filter.Tendsto (fun n => ↑(a n) / ↑n) Filter.atTop Filter.atTop → HasSum (fun n => ↑(a n) / 2 ^ a n) s → Irrational s

irrationalityN/A2 sources

declared openerdos:261

Erdős problem 261

number theoryN/A

declared open · formalizederdos:263

sorry ↔ Erdos263.IsIrrationalitySequence fun n => 2 ^ 2 ^ n

irrationalityN/A3 sources

declared open · formalizederdos:264

¬Erdos264.IsIrrationalitySequence fun x => 2 ^ x

irrationalityN/A4 sources

declared openerdos:265

Erdős problem 265

irrationalityN/A

declared open · formalizederdos:267

n_1 < n_2 < \cdots

with

n_{k+1}/n_k \ge c > 1

, must

\sum_k 1/F_{n_k}

be irrational? The proposed proof closes the range

1 < c < 2

left open by earlier criteria.

irrationalityN/A4 sources

declared open · formalizederdos:269

∀ (P : Set ℕ), (∀ p ∈ P, Nat.Prime p) → P.Infinite → Irrational (Erdos269.series P)

irrationalityN/A2 sources

declared openerdos:271

Erdős problem 271

additive combinatoricsarithmetic progressionsA005487

declared open · formalizederdos:272

Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos272.maxArithInterCard N)) sorry

additive combinatoricsarithmetic progressionspossible2 sources

declared open · formalizederdos:273

True ↔ ∃ c, ∀ (i : c.ι), ∃ p, Nat.Prime p ∧ 5 ≤ p ∧ c.moduli i = Ideal.span {↑(p - 1)}

number theorycovering systemsN/A2 sources

declared open · formalizederdos:274

sorry ↔ ∀ (G : Type u_1) [inst : Group G], 1 < ENat.card G → ∀ (ι : Type u_2) [inst_1 : Fintype ι] (P : Erdos274.Group.ExactCovering G ι), 1 < Fintype.card ι → ∃ i j, i ≠ j ∧ Cardinal.mk ↥(P.parts i) = Cardinal.mk ↥(P.parts j)

group theorycovering systemsN/A3 sources

declared open · formalizederdos:276

True ↔ ∃ a, Erdos276.IsLucasSequence a ∧ (∀ (k : ℕ), (a k).Composite) ∧ ∀ n > 1, ∃ k, n.gcd (a k) = 1

number theorycovering systemsN/A2 sources

declared openerdos:278

Erdős problem 278

number theorycovering systemsN/A

declared open · formalizederdos:279

True ↔ ∀ k ≥ 3, ∃ a N, (∀ (p : ℕ), Nat.Prime p → a p < p) ∧ ∀ n ≥ N, ∃ p, ∃ t ≥ k, Nat.Prime p ∧ n = a p + t * p

number theorycovering systemsprimesN/A2 sources

declared open · formalizederdos:282

∀ {x : ℚ}, x ∈ Set.Ioo 0 1 → Odd x.den → Erdos282.greedyUnitFractionRem {n | Odd n} x =ᶠ[Filter.atTop] 0

number theoryunit fractionsN/A2 sources

declared open · formalizederdos:288

sorry ↔ {I | ∀ (j : Fin 2), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite

number theoryunit fractionsN/A3 sources

declared open · formalizederdos:289

True ↔ ∀ᶠ (k : ℕ) in Filter.atTop, ∃ I, (∀ (i : Fin k), (I i).1 < (I i).2) ∧ (∀ (i j : Fin k), i ≠ j → (I i).2 < (I j).1 ∨ (I j).2 < (I i).1) ∧ ∑ i, ∑ n ∈ Finset.Icc (I i).1 (I i).2, (↑n)⁻¹ = 1

number theoryunit fractionsN/A2 sources

declared open · formalizederdos:291

Erdos291.L 1 = 1 ∧ Erdos291.L 2 = 2 ∧ Erdos291.L 3 = 6 ∧ Erdos291.L 4 = 12

number theoryunit fractionsA1105663 sources

declared openerdos:293

Erdős problem 293

number theoryunit fractionspossible

declared open · formalizederdos:295

sorry ↔ Filter.Tendsto (fun N => ↑(Erdos295.k N) - (Real.exp 1 - 1) * ↑N) Filter.atTop Filter.atTop

number theoryunit fractionsA1928812 sources

declared openerdos:301

Erdős problem 301

number theoryunit fractionsA390394

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 open · formalizederdos:304

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

number theoryunit fractionsA097847 · A0978492 sources

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 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 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

Problems · Vela Mathematics Program

Problem ledger

Search problems.science