Vela Mathematics Program: problems

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316 problems · 1/7Number SourcesGraph view

declared open · formalized · $500erdos:1

∃ C > 0, ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → N ≠ 0 → C * 2 ^ A.card < ↑N

number theoryadditive combinatoricsA2766612 sources

declared open · formalized · $5000erdos:3

True ↔ ∀ (A : Set ℕ), (¬Summable fun a => 1 / ↑↑a) → ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength ↑k

number theoryadditive combinatoricsarithmetic progressionsA003002 · A003003 · A003004 · A0030052 sources

declared openerdos:5

Erdős problem 5

number theoryprimesA001223

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 · formalizederdos:12

Let

A \subset \mathbb{N}

be infinite with no distinct

a, b, c \in A

such that

a \mid (b + c)

with

b, c > a

. Can

|A \cap [1, N]|/\sqrt{N}

have positive lower limit? Must every such

A

fall below

N^{1-c}

infinitely often?

number theoryN/A4 sources

declared open · formalizederdos:14

True ↔ ∀ (A : Set ℕ), ∀ ε > 0, Erdos14.almostSquareRoot ε =O[Filter.atTop] Erdos14.nonUniqueSumCount A

number theorysidon setsadditive combinatoricsA143824 · possible2 sources

declared open · formalizederdos:15

True ↔ Summable fun k => (-1) ^ (k + 1) * (↑k + 1) / ↑(Nat.nth Nat.Prime k)

number theoryprimesN/A2 sources

declared open · formalizederdos:17

sorry ↔ {p | Erdos17.IsClusterPrime p}.Infinite

number theoryprimesA0381332 sources

declared open · formalizederdos:18

∀ᶠ (n : ℕ) in Filter.atTop, Erdos18.practicalH n.factorial < n

number theorydivisorsfactorialsA0051532 sources

declared open · formalizederdos:25

True ↔ ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), (∀ (i : ℕ), 0 < seq_n i) → StrictMono seq_n → ∃ d, {x | ∀ (i : ℕ), ↑x < ↑(seq_n i) ∨ ¬↑x ≡ seq_a i [ZMOD ↑(seq_n i)]}.HasLogDensity d

number theoryN/A3 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 · formalized · $1000erdos:30

True ↔ ∀ ε > 0, (fun N => ↑(Erdos30.h N) - √↑N) =O[Filter.atTop] fun N => ↑N ^ ε

number theorysidon setsadditive combinatoricsA143824 · A227590 · A0030222 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 · formalizederdos:36

Erdos36.M 5 = 3

number theoryadditive combinatoricsA393584 · possible3 sources

declared open · formalized · $500erdos:39

True ↔ ∃ A, A.Infinite ∧ IsSidon A ∧ ∀ ε > 0, (fun x => ↑x ^ (1 / 2 - ε)) =O[Filter.atTop] fun N => ↑(Set.Icc 1 N ∩ A).ncard

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalized · $500erdos:40

Erdos40.Erdos40ForSet sorry

number theoryadditive basisN/A2 sources

declared open · formalized · $500erdos:41

∀ (A : Set ℕ), Erdos41.NtupleCondition A 3 → A.Infinite → Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / ↑N ^ (1 / 3)) Filter.atTop = 0

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalizederdos:44

True ↔ ∀ N ≥ 1, ∀ A ⊆ Finset.Icc 1 N, IsSidon ↑A → ∀ ε > 0, ∃ M > N, ∃ B ⊆ Finset.Icc (N + 1) M, IsSidon (↑A ∪ ↑B) ∧ (1 - ε) * √↑M ≤ ↑(A ∪ B).card

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalized · $250erdos:50

sorry ↔ ∀ (f : ℝ → ℝ), Erdos50.IsDistributionOfPhiRatio f → ¬∃ x ∈ Set.Icc 0 1, ∃ y > 0, HasDerivWithinAt f y (Set.Icc 0 1) x

number theoryN/A2 sources

declared open · formalizederdos:51

True ↔ ∃ A n, A.Infinite ∧ (∀ (a : ↑A), IsLeast (Nat.totient ⁻¹' {↑a}) (n a)) ∧ Filter.Tendsto (fun a => ↑(n a) / ↑↑a) Filter.atTop Filter.atTop

number theoryA002202 · A0141973 sources

declared open · formalized · $250erdos:52

True ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ C, 0 < C ∧ ∀ (A : Finset ℤ), max ↑(A + A).card ↑(A * A).card ≥ C * ↑A.card ^ (2 - ε)

number theoryadditive combinatoricsA2639963 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:68

True ↔ Irrational (∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1))

number theoryirrationalityA3313732 sources

declared openerdos:122

Erdős problem 122

number theoryN/A

declared open · formalizederdos:124

∀ {D : Finset ℕ}, (∀ d ∈ D, 3 ≤ d) → (∀ᶠ (n : ℕ) in Filter.atTop, n ∈ ∑ d ∈ D, Erdos124.sumsOfDistinctPowers d 0) → 1 ≤ ∑ d ∈ D, (↑d - 1)⁻¹

number theorybase representationsN/A4 sources

declared open · formalized · $250erdos:126

True ↔ ∀ (f : ℕ → ℕ), Erdos126.IsMaximalAddFactorsCard f → Filter.Tendsto (fun n => ↑(f n) / Real.log ↑n) Filter.atTop Filter.atTop

number theorypossible2 sources

declared openerdos:131

Let

F(N)

be the maximal size of

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

such that no

a\in A

divides the sum of any nonempty subset of

A\setminus\{a\}

. Estimate

F(N)

. The lower bound

F(N)\gg N^{1/5}

is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave

F(N)\leq N^{1/4+o(1)}

. The claimed new result is the matching upper bound

F(N)\leq N^{1/5+o(1)}

, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence

F(N)=N^{1/5+o(1)}

number theoryA0680632 sources

declared open · formalizederdos:137

sorry ↔ ∀ k ≥ 3, ∀ (n : ℕ), ¬(∏ x ∈ Finset.Ioc n (n + k), x).Powerful

number theoryN/A2 sources

declared open · formalizederdos:145

True ↔ ∀ α ≥ 0, ∃ β, Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop (nhds β)

number theoryA0051172 sources

declared openerdos:148

Erdős problem 148

number theoryunit fractionsA076393 · A006585

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: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 · 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:261

Erdős problem 261

number theoryN/A

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:276

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

number theorycovering systemsN/A2 sources

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