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declared open · formalizederdos:9
1 ∈ Erdos9.Erdos9A
number theoryadditive basisprimesA0062862 sources
declared open · formalizederdos:10
True ↔ ∃ k, Erdos10.sumPrimeAndTwoPows k = Set.univ \ {0, 1}
number theoryadditive basisprimesA3870532 sources
declared open · formalizederdos:11
∀ (n : ℕ), Odd n → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l
number theoryadditive basisA001220 · A3775873 sources
declared open · formalized · $500erdos:28
∀ (A : Set ℕ), (A + A)ᶜ.Finite → Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep A n)) Filter.atTop = ⊤
number theoryadditive basisN/A2 sources
declared open · formalizederdos:32
True ↔ ∃ A, Erdos32.IsAdditiveComplementToPrimes A ∧ (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card) =o[Filter.atTop] fun N => Real.log ↑N ^ 2
number theoryadditive basisN/A2 sources
declared open · formalizederdos:33
⨅ A, Filter.limsup (fun N => ↑(↑A ∩ Set.Icc 1 N).ncard / ↑√↑N) Filter.atTop = sorry
number theoryadditive basisN/A2 sources
declared open · formalized · $500erdos:40
Erdos40.Erdos40ForSet sorry
number theoryadditive basisN/A2 sources
declared open · formalized · $500erdos:66
True ↔ ∃ A c, c ≠ 0 ∧ Filter.Tendsto (fun n => ↑(AdditiveCombinatorics.sumRep A n) / Real.log ↑n) Filter.atTop (nhds c)
number theoryadditive basisN/A2 sources
declared open · formalizederdos:326
Does there exist A={a1<a2<⋯ }⊂NA=\{a_1<a_2<\cdots\}\subset \mathbb{N} which is a minimal basis of order 22 (every large integer is the sum of 22 elements from AA, and no proper subset of AA has this property) such that lim⁡k→∞ak/k2=c\lim_{k\to \infty}a_k/k^2=c for some c≠0c\neq 0? A claimed construction gives a minimal basis with A(x)=Cx+O(1)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:336
If h(r)h(r) is the maximal finite exact order attainable by an additive basis of order at most rr, what is lim⁡r→∞h(r)/r2\lim_{r \to \infty} h(r)/r^2? The candidate proof identifies the sharp limit 1/31/3.
number theoryadditive basispossible2 sources
declared openerdos:338
Erdős problem 338
number theoryadditive basisN/A
declared openerdos:870
Let k≥3k\geq 3 and AA be an additive basis of order kk. Does there exist a constant c=c(k)>0c=c(k)>0 such that if r(n)≥clog⁡nr(n)\geq c\log n for all large nn (where r(n)r(n) counts representations of nn as a sum of at most kk elements of AA) then AA must contain a minimal basis of order kk? The claimed answer is no, for every k≥3k\geq 3.
number theoryadditive basisN/A3 sources
declared open · formalizederdos:881
True ↔ ∀ (k : ℕ) (A : Set ℕ), Erdos881.IsMinimalAsymptoticAddBasisOfOrder k A → ∃ B ⊆ A, B.Infinite ∧ (A \ B).IsAsymptoticAddBasisOfOrder (k + 1)
number theoryadditive basisN/A2 sources
declared open · formalizederdos:1145
True ↔ Erdos1145.Erdos1145Prop
additive combinatoricsadditive basisN/A2 sources
declared open · formalizederdos:1192
True ↔ ∀ r ≥ 2, ∃ A, (∀ᶠ (n : ℕ) in Filter.atTop, Erdos1192.f_r A r n > 0) ∧ (fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A r n) ^ 2) =O[Filter.atTop] fun x => ↑x
additive combinatoricsadditive basispossible2 sources
declared openerdos:1194
Erdős problem 1194
additive combinatoricsadditive basissidon setspossible2 sources

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