962.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 962

References:

erdosproblems.com/962
[Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.
[Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
Tang
Tao

Imports (1)

FormalConjecturesUtil

def Erdos962.Erdos962Prop Erdos962Prop n k : there exists $m \le n$ such that each of $m+1, \ldots, m+k$ has a prime divisor strictly larger than $k$ line 41 source
Source documentation
Erdos962Prop n k : there exists $m \le n$ such that each of $m+1, \ldots, m+k$ has a prime divisor strictly larger than $k$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, LE.le, Membership.mem, Set, Set.Icc, OfNat.ofNat, Nat.Prime, LT.lt, Dvd.dvd, HAdd.hAdd
8 instances, coercions and auxiliary constants left out.
```
def Erdos962Prop (n k : ℕ) : Prop :=
…
```
Mentions: Erdos962.k
def Erdos962.k Let $k(n)$ be the maximal $k$ such that there exists $m \le n$ with $m+1, \ldots, m+k$ each divisible by a prime $> k$ line 49 source
Source documentation
Let $k(n)$ be the maximal $k$ such that there exists $m \le n$ with $m+1, \ldots, m+k$ each divisible by a prime $> k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.findGreatest, Erdos962.Erdos962Prop, Classical.propDecidable
```
noncomputable def k (n : ℕ) : ℕ :=
…
```
theorem Erdos962.erdos_962 Main conjecture: $\log k(n) \le (\log n)^{(1/2 + o(1))}$ sorry line 59 source
Source documentation
Main conjecture:
$\log k(n) \le (\log n)^{(1/2 + o(1))}$

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, abs, Filter.atTop, LE.le, Real.log, Erdos962.k, Real.rpow, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_962 :
    answer(sorry) ↔
      ∃ ε : ℕ → ℝ,
        (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
        ∀ᶠ n : ℕ in atTop,
          log (k n : ℝ)
            ≤ rpow (log n) ((1 : ℝ) / 2 + ε n) := by
…
```
Mentions: Erdos962.k
theorem Erdos962.erdos_962.variants.tang_lower_bound Tang's lower bound [Tang]: $\log k(n) \ge (1/\sqrt{2} - o(1)) * \sqrt{\log n * \log \log n}$ sorry line 74 source
Source documentation
Tang's lower bound [Tang]:
$\log k(n) \ge (1/\sqrt{2} - o(1)) * \sqrt{\log n * \log \log n}$

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, abs, Filter.atTop, LE.le, HMul.hMul, HSub.hSub, HDiv.hDiv, Real.sqrt, Real.log, Erdos962.k
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_962.variants.tang_lower_bound :
    ∃ ε : ℕ → ℝ,
      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
      ∀ᶠ n : ℕ in atTop,
          (1 / sqrt 2 - ε n) *
              sqrt (log n * log (log n))
            ≤ log (k n : ℝ) := by
…
```
Mentions: Erdos962.k
theorem Erdos962.erdos_962.variants.tao_upper_bound Tao's upper bound [Tao]: $k(n) \le (1 + o(1)) * n^{1/2}$ sorry line 89 source
Source documentation
Tao's upper bound [Tao]:
$k(n) \le (1 + o(1)) * n^{1/2}$

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, LT.lt, abs, Filter.atTop, LE.le, Erdos962.k, HMul.hMul, HAdd.hAdd, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_962.variants.tao_upper_bound :
    ∃ ε : ℕ → ℝ,
      (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
      ∀ᶠ n : ℕ in atTop,
          (k n : ℝ) ≤ (1 + ε n) * sqrt n := by
…
```
Mentions: Erdos962.k