86766.lean`

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L4DEL: Lean4 Domain Explanation Language

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

9 declarations, 3 contain sorry, 0 declared as axioms

Smallest $r$ such that (concatenation of $n$, $r$ times) $\cdot 10 + 1$ is prime

$a(n)$ is the smallest $r$ where (concatenation of $n$, $r$ times with itself) $\cdot 10 + 1$ is a prime, or $0$ if no such number exists. The number resulting from concatenating $n$, $r$ times, is $n \cdot \sum_{i=0}^{r-1} (10^d)^i$, where $d$ is the number of digits of $n$.

References:

A086766

Imports (1)

FormalConjecturesUtil

def OeisA86766.a Sequence $a(n)$ is the smallest $r > 0$ such that the concatenation of $n$, $r$ times with itself, multiplied by $10$ plus $1$, is prime, or $0$ if no such prime exists line 38 source
Source documentation
Sequence $a(n)$ is the smallest $r > 0$ such that the concatenation of $n$, $r$ times with itself, multiplied by $10$ plus $1$, is prime, or $0$ if no such prime exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, List.length, Nat.digits, HPow.hPow, HMul.hMul, Finset.sum, Finset.range, HAdd.hAdd, InfSet.sInf, Set.ofPred, And, LT.lt, Nat.Prime
13 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA86766.a_0 Value of the sequence a at 0 line 49 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA86766.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA86766.a
theorem OeisA86766.a_1 Value of the sequence a at 1 line 53 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA86766.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, Finset.sum, Finset.range, HPow.hPow, List.length, Nat.digits, And.intro, Membership.mem, Set, lowerBounds, id, Decidable.byContradiction, Not, of_decide_eq_true and 40 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.a_2 Value of the sequence a at 2 line 73 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA86766.a, OfNat.ofNat
Proof uses: List.length, Nat.digits, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, Finset.sum, Finset.range, HPow.hPow, And.intro, Membership.mem and 65 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 3 := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.a_3 Value of the sequence a at 3 line 103 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA86766.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, Finset.sum, Finset.range, HPow.hPow, List.length, Nat.digits, And.intro, Membership.mem, Set, lowerBounds, id, Decidable.byContradiction, Not, of_decide_eq_true and 40 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.a_4 Value of the sequence a at 4 line 123 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA86766.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, Finset.sum, Finset.range, HPow.hPow, List.length, Nat.digits, And.intro, Membership.mem, Set, lowerBounds, id, Decidable.byContradiction, Not, of_decide_eq_true and 40 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.conjecture1 What is the smallest integer $m > 1$ such that $a(10^m)$ is nonzero? sorry line 147 source
Source documentation
What is the smallest integer $m > 1$ such that $a(10^m)$ is nonzero?
- Farideh Firoozbakht, Jan 07 2015
depends on sorry what this means
Statement uses: Eq, Option, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, dite, Exists, And, LT.lt, Ne, OeisA86766.a, HPow.hPow, Classical.propDecidable, Option.some, InfSet.sInf and 3 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 :
    answer(sorry) =
      if h : ∃ m, 1 < m ∧ a (10 ^ m) ≠ 0 then
        some (sInf {m | 1 < m ∧ a (10 ^ m) ≠ 0})
      else
        none := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.conjecture2 Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero sorry line 161 source
Source documentation
Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero.
- Farideh Firoozbakht, Jan 07 2015
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, HPow.hPow, OeisA86766.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (n : ℕ) (hn : 0 < n) (h : ∀ m : ℕ, n ≠ 10 ^ m) : a n ≠ 0 := by
…
```
Mentions: OeisA86766.a
theorem OeisA86766.conjecture3 What is the smallest odd prime $p$ such that $(10^{p^2}-1)/(10^p-1)$ is a prime number (and $a(10^{p-1})$ could be nonzero)? sorry line 171 source
Source documentation
What is the smallest odd prime $p$ such that $(10^{p^2}-1)/(10^p-1)$ is a prime number (and $a(10^{p-1})$ could be nonzero)?
- Farideh Firoozbakht, Jan 07 2015
depends on sorry what this means
Statement uses: Eq, Option, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, dite, Exists, And, Nat.Prime, LT.lt, HDiv.hDiv, HSub.hSub, HPow.hPow, Classical.propDecidable, Option.some and 4 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 :
    answer(sorry) =
      if h : ∃ p, p.Prime ∧ 2 < p ∧ ((10 ^ (p ^ 2) - 1) / (10 ^ p - 1)).Prime then
        some (sInf {p | p.Prime ∧ 2 < p ∧ ((10 ^ (p ^ 2) - 1) / (10 ^ p - 1)).Prime})
      else
        none := by
…
```