377.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 377

Reference: erdosproblems.com/377

Imports (1)

FormalConjecturesUtil

abbrev Erdos377.sumInvPrimesNotDvdCentralBinom The sum of the inverses of all primes smaller than $n$, which don't divide the central binom coefficient line 38 source
Source documentation
The sum of the inverses of all primes smaller than $n$, which don't divide the central binom coefficient.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.sum, Finset.filter, Nat.Prime, Finset.Icc, OfNat.ofNat, ite, Dvd.dvd, Nat.centralBinom, HDiv.hDiv
16 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev sumInvPrimesNotDvdCentralBinom (n : ℕ) : ℝ :=
…
```
theorem Erdos377.erdos_377 Is there some absolute constant $C > 0$ such that $$ \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C $$ for all $n$? sorry line 49 source
Source documentation
Is there some absolute constant $C > 0$ such that $$ \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C $$ for all $n$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LE.le, Erdos377.sumInvPrimesNotDvdCentralBinom
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_377 : answer(sorry) ↔
    ∃ C > (0 : ℝ), ∀ (n : ℕ), sumInvPrimesNotDvdCentralBinom n ≤ C := by
…
```
Mentions: Erdos377.sumInvPrimesNotDvdCentralBinom
theorem Erdos377.erdos_377.variants.limit.i Documented declaration: i sorry line 70 source
Source documentation
Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then $$ \lim_{x\to\infty} \frac{1}{x}\sum_{n\leq x} f(n) = \gamma_0 $$
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.

depends on sorry what this means
Statement uses: Real, Eq, tsum, Nat, HDiv.hDiv, Real.log, HAdd.hAdd, OfNat.ofNat, HPow.hPow, SummationFilter.unconditional, Filter.Tendsto, HMul.hMul, Finset.sum, Finset.Icc, Erdos377.sumInvPrimesNotDvdCentralBinom, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_377.variants.limit.i (γ₀ : ℝ)
    (hγ₀ : γ₀ = ∑' (k : ℕ), (k + 2 : ℝ).log / 2 ^ (k + 2)) :
    Tendsto (fun (x : ℕ) => (1 : ℝ) / x * ∑ n ∈ Finset.Icc 1 x, sumInvPrimesNotDvdCentralBinom n)
      atTop (𝓝 γ₀) := by
…
```
Mentions: Erdos377.sumInvPrimesNotDvdCentralBinom
theorem Erdos377.erdos_377.variants.limit.ii Documented declaration: ii sorry line 93 source
Source documentation
Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then $$ \lim_{x\to\infty} \frac{1}{x}\sum_{n\leq x} f(n)^2 = \gamma_0^2 $$
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.

depends on sorry what this means
Statement uses: Real, Eq, tsum, Nat, HDiv.hDiv, Real.log, HAdd.hAdd, OfNat.ofNat, HPow.hPow, SummationFilter.unconditional, Filter.Tendsto, HMul.hMul, Finset.sum, Finset.Icc, Erdos377.sumInvPrimesNotDvdCentralBinom, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_377.variants.limit.ii (γ₀ : ℝ)
    (hγ₀ : γ₀ = ∑' (k : ℕ), (k + 2 : ℝ).log / 2 ^ (k + 2)) :
    Tendsto (fun (x : ℕ) =>
      (1 : ℝ) / x * ∑ n ∈ Finset.Icc 1 x, sumInvPrimesNotDvdCentralBinom n ^ 2)
      atTop (𝓝 (γ₀ ^ 2)) := by
…
```
Mentions: Erdos377.sumInvPrimesNotDvdCentralBinom
theorem Erdos377.erdos_377.variants.ae Documented declaration: ae sorry line 115 source
Source documentation
Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then for almost all integers $f(m) = \gamma_0 + o(1)$, i.e. for every $\varepsilon > 0$ the set of $n$ with $|f(n) - \gamma_0| < \varepsilon$ has natural density one.
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.

depends on sorry what this means
Statement uses: Real, Eq, tsum, Nat, HDiv.hDiv, Real.log, HAdd.hAdd, OfNat.ofNat, HPow.hPow, SummationFilter.unconditional, GT.gt, Set.HasDensity, Set.ofPred, LT.lt, abs, HSub.hSub, Erdos377.sumInvPrimesNotDvdCentralBinom, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_377.variants.ae (γ₀ : ℝ) (hγ₀ : γ₀ = ∑' (k : ℕ), (k + 2 : ℝ).log / 2 ^ (k + 2)) :
    ∀ ε > (0 : ℝ), {n : ℕ | |sumInvPrimesNotDvdCentralBinom n - γ₀| < ε}.HasDensity 1 := by
…
```
Mentions: Erdos377.sumInvPrimesNotDvdCentralBinom
theorem Erdos377.erdos_377.variants.ub Documented declaration: ub sorry line 132 source
Source documentation
Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ then there is some constant $c < 1$ such that for all large $n$ $$ f(n) \leq c \log\log n. $$
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos377.sumInvPrimesNotDvdCentralBinom, HMul.hMul, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_377.variants.ub : ∃ c < (1 : ℝ),
      ∀ᶠ n in atTop, sumInvPrimesNotDvdCentralBinom n ≤ c * (n : ℝ).log.log := by
…
```
Mentions: c, Erdos377.sumInvPrimesNotDvdCentralBinom