38098.lean`

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L4DEL: Lean4 Domain Explanation Language

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

8 declarations, 2 contain sorry, 0 declared as axioms

Number of primes $< n^3$

Number of primes strictly less than $n^3$.

References:

A038098

Imports (1)

FormalConjecturesUtil

def OeisA38098.a Number of primes strictly less than $n^3$ line 34 source
Source documentation
Number of primes strictly less than $n^3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Nat.primesBelow, HPow.hPow, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := (Nat.primesBelow (n ^ 3)).card
…
```
theorem OeisA38098.a_1 Value of the sequence a at 1 line 38 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38098.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by
…
```
Mentions: OeisA38098.a
theorem OeisA38098.a_2 Value of the sequence a at 2 line 43 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38098.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 4 := by
…
```
Mentions: OeisA38098.a
theorem OeisA38098.a_3 Value of the sequence a at 3 line 48 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38098.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 9 := by
…
```
Mentions: OeisA38098.a
theorem OeisA38098.a_4 Value of the sequence a at 4 line 53 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38098.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 18 := by
…
```
Mentions: OeisA38098.a
theorem OeisA38098.a_5 Value of the sequence a at 5 line 58 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38098.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 30 := by
…
```
Mentions: OeisA38098.a
theorem OeisA38098.conjecture1 Conjecture (i): for any integer $k > 2$, the sequence $\pi(n^k)/n^k$ ($n = 2, 3, \ldots$) is strictly decreasing, where $\pi(x)$ denotes the number of primes not exceeding $x$ sorry line 67 source
Source documentation
Conjecture (i): for any integer $k > 2$, the sequence $\pi(n^k)/n^k$ ($n = 2, 3, \ldots$) is strictly decreasing, where $\pi(x)$ denotes the number of primes not exceeding $x$.
- Zhi-Wei Sun, Oct 17 2015
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, LE.le, Rat, HDiv.hDiv, Nat.primeCounting, HPow.hPow, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (k : ℕ) (hk : 2 < k) (n : ℕ) (hn : 2 ≤ n) :
    (Nat.primeCounting ((n + 1) ^ k) : ℚ) / ((n + 1) ^ k : ℚ) <
    (Nat.primeCounting (n ^ k) : ℚ) / (n ^ k : ℚ) := by
…
```
theorem OeisA38098.conjecture2 Conjecture (ii): all the numbers $\pi(n^2)/n^2$ ($n = 1, 2, 3, \ldots$) are pairwise distinct sorry line 78 source
Source documentation
Conjecture (ii): all the numbers $\pi(n^2)/n^2$ ($n = 1, 2, 3, \ldots$) are pairwise distinct. Moreover, we have $\pi(n^2)/n^2 > \pi((n+1)^2)/(n+1)^2$ for all $n > 15646$.
- Zhi-Wei Sun, Oct 17 2015
depends on sorry what this means
Statement uses: And, Nat, LE.le, OfNat.ofNat, Eq, Rat, HDiv.hDiv, Nat.primeCounting, HPow.hPow, LT.lt, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
    (∀ m n : ℕ, 1 ≤ m → 1 ≤ n →
      (Nat.primeCounting (m ^ 2) : ℚ) / (m ^ 2 : ℚ) =
      (Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ) → m = n) ∧
    (∀ n : ℕ, 15646 < n →
      (Nat.primeCounting ((n + 1) ^ 2) : ℚ) / ((n + 1) ^ 2 : ℚ) <
      (Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ)) := by
…
```