Firoozbakht.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Firoozbakht's conjecture

References:

Imports (1)

FormalConjecturesUtil

def Firoozbakht.firoozbakhtSeq The sequence of $\sqrt[n]{p_n}$ where $p_n$ is the n:th prime number line 37 source
Source documentation
The sequence of $\sqrt[n]{p_n}$ where $p_n$ is the n:th prime number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HPow.hPow, Nat.nth, Prime, HDiv.hDiv, OfNat.ofNat, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
noncomputable def firoozbakhtSeq (n : ℕ) : ℝ :=
…
```
theorem Firoozbakht.firoozbakht_conjecture Firoozbakht's conjecture The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$ sorry line 45 source
Source documentation
Firoozbakht's conjecture The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$.

depends on sorry what this means
Statement uses: Nat, LT.lt, Real, Firoozbakht.firoozbakhtSeq, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem firoozbakht_conjecture (n : ℕ) :
    firoozbakhtSeq (n+1) < firoozbakhtSeq n := by
…
```
Mentions: Firoozbakht.firoozbakhtSeq
theorem Firoozbakht.firoozbakht_conjecture_consequence The inequality $p_{n+1}-p_n < (\log p_n)^2-\log p_n$ holds for all $n>4$ sorry line 54 source
Source documentation
The inequality $p_{n+1}-p_n < (\log p_n)^2-\log p_n$ holds for all $n>4$. A consequence of Firuzbakht's conjecture.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Real, Firoozbakht.firoozbakhtSeq, HAdd.hAdd, HSub.hSub, Nat.nth, Prime, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem firoozbakht_conjecture_consequence (n : ℕ) (hn : 3 < n) (P : type_of% firoozbakht_conjecture) :
    (n+1).nth Prime - n.nth Prime < (log (n.nth Prime))^2 - log (n.nth Prime) := by
…
```
Mentions: Firoozbakht.firoozbakht_conjecture