115366.lean`

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L4DEL: Lean4 Domain Explanation Language

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

6 declarations, 1 contain sorry, 0 declared as axioms

$a(n)$ = the number of values of $k <= 10^n$ such that $\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime

Since $\sqrt{k(k+1)(k+2)(k+3)+1} = k^2 + 3k + 1$, $a(n) = \#\{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} \}.$

References:

A115366

Imports (1)

FormalConjecturesUtil

def OeisA115366.a The primary defining sequence a line 40 source
Source documentation
The primary defining sequence a. $a(n) = \#\{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} \}.$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Nat.Prime, HAdd.hAdd, HPow.hPow, OfNat.ofNat, HMul.hMul, Finset.Icc
12 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA115366.a_0 line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115366.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 OeisA115366.a_1 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115366.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 OeisA115366.a_2 line 53 source
extra axioms: _private.0.OeisA115366.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA115366.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA115366.a_3 line 56 source
extra axioms: _private.0.OeisA115366.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA115366.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA115366.conjecture Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$ where A006880(n) is the number of primes $\le 10^n$ sorry line 63 source
Source documentation
Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$ where A006880(n) is the number of primes $\le 10^n$.

depends on sorry what this means
Statement uses: Exists, Real, And, Filter.Tendsto, Nat, HDiv.hDiv, OeisA115366.a, Nat.primeCounting', HPow.hPow, OfNat.ofNat, Filter.atTop, nhds, LE.le, OfScientific.ofScientific, Bool.true
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 conjecture :
    ∃ L : ℝ,
      Tendsto (fun n : ℕ => (a n : ℝ) / (Nat.primeCounting' (10 ^ n) : ℝ)) atTop (nhds L) ∧
      1.77 ≤ L ∧ L ≤ 1.78 := by
…
```
Mentions: OeisA115366.a