Vela Mathematics Program: problems

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576 problems · 12/12Number SourcesGraph view

declared openerdos:1144

Erdős problem 1144

number theoryprobabilityN/A

declared open · formalizederdos:1146

True ↔ Erdos1146.IsEssentialComponent {k | ∃ m n, k = 2 ^ m * 3 ^ n}

number theorypossible2 sources

declared proved (Lean) · formalizederdos:1148

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1148.Erdos1148Prop n

number theoryA390380 · A3931686 sources

declared provederdos:1149

Erdős problem 1149

number theoryN/A

declared provederdos:1180

Erdős problem 1180

number theoryN/A

declared openerdos:1181

Erdős problem 1181

number theorypossible

declared openerdos:1184

Erdős problem 1184

number theoryprimespossible

declared solvederdos:1187

Erdős problem 1187

number theoryprimesadditive combinatoricsN/A3 sources

declared open · formalizederdos:1188

Estimate the number

F(x)

of minimal distinct covering systems whose moduli all lie in

[1, x]

. The candidate proof gives

\log\log F(x)/\log x \to 1

, i.e.

F(x) = \exp(x^{1+o(1)})

number theorycovering systemspossible4 sources

declared openerdos:1189

For irreducible covering sets of size

k

, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

number theorycovering systemspossible2 sources

declared solved (Lean) · formalizederdos:1190

∀ (ε : ℝ), 0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, scaleL m ^ (-1 - ε) < Erdos1190.eps m ∧ Erdos1190.eps m < scaleL m ^ (-1 + ε)

number theorycovering systemspossible6 sources

declared solvederdos:1195

Erdős problem 1195

number theorypossible3 sources

declared proved (Lean) · formalizederdos:1196

Bounds the weighted sum

\sum 1/(a \log a)

taken over primitive sets of integers (sets where no element divides another).

number theoryprimitive setspossible6 sources

declared openerdos:1200

Erdős problem 1200

number theoryprimespossible

declared open · formalizederdos:1201

Is it true that for every

\epsilon,\eta>0

there exists a

k

such that the density of

n

for which

P(n(n+1)\cdots(n+k))>n^{1-\epsilon}

is at least

1-\eta

, where

P(m)

is the greatest prime divisor of

m

? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.

number theoryprimespossible4 sources

declared solvederdos:1202

Erdős problem 1202

number theoryprimespossible3 sources

declared open · formalizederdos:1203

True ↔ Filter.Tendsto Erdos1203.F Filter.atTop Filter.atTop

number theorypossible2 sources

declared openerdos:1204

Erdős problem 1204

number theoryA008407 · A023193 · A135311 · possible

declared solvederdos:1205

Erdős problem 1205

number theorypossible

declared openerdos:1206

Erdős problem 1206

number theorysidon setspossible

declared open · formalizederdos:1209

False ↔ ∃ f, ∀ (a : ℕ → ℕ), StrictMono a → (∀ (k : ℕ), f k ≤ a k) → (∃ n, ∀ (k : ℕ), Nat.Prime (n + a k)) → {n | ∀ (k : ℕ), Nat.Prime (n + a k)}.Infinite

number theorypossible3 sources

declared open · formalizederdos:1210

True ↔ ∃ C, ∀ (n : ℕ) (A : Finset ℕ), (∀ a ∈ A, 1 ≤ a ∧ a < n) → (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) → ∑ a ∈ A, 1 / (↑n - ↑a) ≤ ∑ p ∈ Finset.range n with Prime p, 1 / ↑p + C

number theorypossible2 sources

declared solvederdos:1211

Erdős problem 1211

number theoryramsey theorypossible

declared open · formalizederdos:1212

∀ {a s : ℕ}, a < s → (∀ (p : ℕ), Nat.Prime p → p ∣ a → s < a + p) → a.gcd s = 1

number theoryprimespossible2 sources

declared proved · formalizederdos:1214

True ↔ ∀ (x y : ℕ), x ≥ 1 → y ≥ 1 → (∀ n ≥ 1, {p | Nat.Prime p ∧ p ∣ x ^ n - 1} = {p | Nat.Prime p ∧ p ∣ y ^ n - 1}) → x = y

number theoryN/A2 sources

declared provederdos:1217

Erdős problem 1217

number theorydivisorsprimitive setsN/A3 sources

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