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declared disproved (Lean) · formalizederdos:16

False ↔ ∃ A B, Erdos16.Erdos16Set = A ∪ B ∧ (∃ a, ∃ d > 0, A = {x | ∃ m, x = a + m * d}) ∧ Erdos16.density_zero B

number theoryadditive basisprimesA0062855 sources

declared disproved (Lean) · formalizederdos:26

A\subset\mathbb{N}

be infinite. Must there exist some

k\geq 1

such that almost all integers have a divisor of the form

a+k

for some

a\in A

? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite

A

such that for every

k\geq 1

the set of multiples of

A+k

has upper density below

0.34

number theorydivisorsN/A6 sources

declared disproved (Lean) · formalizederdos:34

False ↔ ∀ (c : ℝ), 0 < c → ∃ N, ∀ n ≥ N, ∀ (p : Equiv.Perm (Fin n)), ↑(Erdos34.consecutiveSums n p).card < c * ↑n ^ 2

number theoryA389241 · A234813 · A3901875 sources

declared disproved (Lean) · formalized · $10erdos:56

∀ {A : Finset ℕ}, Erdos56.WeaklyDivisible 0 A ↔ A = ∅

number theoryintersecting familyN/A5 sources

declared disproved (Lean) · formalizederdos:125

False ↔ ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).HasPosDensity

number theorybase representationsA3670906 sources

declared disproved (Lean) · formalizederdos:205

False ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos205.IsRepresentable (fun m => Real.log (Real.log ↑m)) n

number theorypossible6 sources

declared disproved (Lean) · formalizederdos:206

False ↔ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioi 0), Erdos206.EventuallyGreedy x

number theoryunit fractionsN/A5 sources

declared disproved (Lean) · formalizederdos:280

False ↔ ∀ (n a : ℕ → ℕ), StrictMono n → (∀ (i : ℕ), 1 ≤ i → a i < n i) → (∃ ε, 0 < ε ∧ ∀ (k : ℕ), 1 ≤ k → ↑(n k) > (1 + ε) * ↑k * Real.log ↑k) → ¬Filter.Tendsto (fun k => ↑(Erdos280.uncoveredCount n a k) / ↑k) Filter.atTop (nhds 0)

number theorycovering systemsN/A5 sources

declared disproved (Lean) · formalizederdos:299

False ↔ ∃ a, StrictMono a ∧ (∀ (n : ℕ), 0 < a n) ∧ (fun n => ↑(a (n + 1)) - ↑(a n)) =O[Filter.atTop] 1 ∧ ∀ (S : Finset ℕ), ∑ i ∈ S, 1 / ↑(a i) ≠ 1

number theoryunit fractionsN/A4 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 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 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 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 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 disproved (Lean) · formalizederdos:363

False ↔ {S | Erdos363.IsValidCollection S}.Finite

number theoryN/A5 sources

declared disproved (Lean) · formalizederdos:397

False ↔ {(M, N) | Disjoint M N ∧ ∏ i ∈ M, i.centralBinom = ∏ j ∈ N, j.centralBinom}.Finite

number theorybinomial coefficientsN/A7 sources

declared disproved (Lean) · formalizederdos:399

False ↔ ¬∃ n x y k, 1 < x * y ∧ 2 < k ∧ (n.factorial = x ^ k + y ^ k ∨ n.factorial + y ^ k = x ^ k)

number theoryfactorialsN/A5 sources

declared disproved (Lean) · formalizederdos:429

False ↔ ∃ f, Filter.Tendsto f Filter.atTop Filter.atTop ∧ ∀ (A : Set ℕ), A.Infinite → (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) → (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)

number theoryN/A5 sources

declared disproved (Lean) · formalizederdos:453

False ↔ Erdos453.EventuallyHasPrimeWitness

number theoryN/A5 sources

declared disproved (Lean) · formalizederdos:537

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ↑A.card ≥ ε * ↑N → ∃ a₁ ∈ A, ∃ a₂ ∈ A, ∃ a₃ ∈ A, ∃ p₁ p₂ p₃, Nat.Prime p₁ ∧ Nat.Prime p₂ ∧ Nat.Prime p₃ ∧ p₁ ≠ p₂ ∧ p₁ ≠ p₃ ∧ p₂ ≠ p₃ ∧ a₁ * p₁ = a₂ * p₂ ∧ a₂ * p₂ = a₃ * p₃

number theorypossible5 sources

declared disproved (Lean) · formalizederdos:649

False ↔ ∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → ∃ n, n.maxPrimeFac = p ∧ (n + 1).maxPrimeFac = q

number theorypossible5 sources

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 disproved (Lean) · formalizederdos:845

False ↔ ∀ (C : ℝ), 0 < C → have f := fun x => match x with | (k, l) => 2 ^ k * 3 ^ l; {x | ∃ B, ∃ (h : B.Nonempty) (_ : ↑(B.sup f) ≤ C * ↑(B.inf' h f)), ∑ x ∈ B, f x = x}.HasDensity 0

number theoryN/A5 sources

declared disproved (Lean) · formalizederdos:871

False ↔ ∀ (A : Set ℕ), ((∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ A, ∃ b ∈ A, a + b = n) ∧ ∀ (t : ℕ), ∀ᶠ (n : ℕ) in Filter.atTop, ∃ pairs, pairs.card ≥ t ∧ ∀ p ∈ pairs, p.1 ∈ A ∧ p.2 ∈ A ∧ p.1 + p.2 = n ∧ p.1 ≤ p.2) → ∃ B C, (∀ (x : ℕ), x ∈ A ↔ x ∈ B ∨ x ∈ C) ∧ Disjoint B C ∧ (∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ B, ∃ b ∈ B, a + b = n) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ C, ∃ b ∈ C, a + b = n

number theoryadditive basisN/A6 sources

declared disproved (Lean) · formalizederdos:884

False ↔ Erdos884.Erdos884Prop

number theorydivisorsN/A3 sources

declared disproved (Lean) · formalizederdos:897

False ↔ ∀ (f : ℕ → ℝ), (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) → Filter.limsup (fun x => match x with | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k))) (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) = ⊤ → Filter.limsup (fun n => (↑(f (n + 1)) - ↑(f n)) / ↑(Real.log ↑n)) Filter.atTop = ⊤

number theoryN/A6 sources

declared disproved (Lean) · formalizederdos:967

False ↔ ∀ (a : ℕ → ℕ), StrictMono a → 1 < a 0 → (Summable fun k => 1 / ↑(a k)) → ∀ (t : ℝ), 1 + ∑' (k : ℕ), Erdos967.summand t (a k) ≠ 0

number theoryanalysisN/A5 sources

declared disproved (Lean) · £10erdos:1134

Erdős problem 1134

number theoryA1856613 sources

declared disproved (Lean) · formalizederdos:1138

False ↔ ∀ C > 1, Asymptotics.IsEquivalent Erdos1138.snd_gt_half_fst (Erdos1138.primeCount_Ioc_mul_const C) fun x => match x with | (x, y) => C * ↑(Erdos1138.sup_primeGap x) / Real.log y

number theoryprimespossible6 sources

declared disproved (Lean) · formalizederdos:1141

False ↔ Infinite ↑{n | Erdos1141.Erdos1141Prop n}

number theoryprimesA2145836 sources

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