78729.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Least $k > 0$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime

The sequence $a(n)$ is the least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, if such $k$ exists; otherwise $a(n) = 0$.

References:

A078729

Imports (1)

FormalConjecturesUtil

def OeisA78729.a Least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, or 0 if no such $k$ exists line 37 source
Source documentation
Least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, or 0 if no such $k$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: dite, Exists, And, LT.lt, OfNat.ofNat, Nat.Prime, HAdd.hAdd, Finset.prod, Finset.range, Classical.propDecidable, Nat.find, Not
8 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA78729.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

extra axioms: _private.0.OeisA78729.a_1._native.decide.ax_1, _private.0.OeisA78729.a_1._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78729.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, Finset.prod, Singleton.singleton, Finset, Classical.propDecidable, Finset.range, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff and 9 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA78729.a
theorem OeisA78729.a_2 Value of the sequence a at 2 line 58 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA78729.a_2._native.decide.ax_1, _private.0.OeisA78729.a_2._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78729.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, Finset.prod, Finset.range, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
13 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by
…
```
Mentions: OeisA78729.a
theorem OeisA78729.a_3 Value of the sequence a at 3 line 68 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA78729.a_3._native.decide.ax_1, _private.0.OeisA78729.a_3._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78729.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, Finset.prod, Finset.range, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
13 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2 := by
…
```
Mentions: OeisA78729.a
theorem OeisA78729.a_5 Value of the sequence a at 5 line 78 source
Source documentation
Value of the sequence a at 5.

extra axioms: _private.0.OeisA78729.a_5._native.decide.ax_1, _private.0.OeisA78729.a_5._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78729.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, Finset.prod, Finset.range, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
13 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 2 := by
…
```
Mentions: OeisA78729.a
theorem OeisA78729.a_6 Value of the sequence a at 6 line 88 source
Source documentation
Value of the sequence a at 6.

extra axioms: _private.0.OeisA78729.a_6._native.decide.ax_1, _private.0.OeisA78729.a_6._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78729.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, Finset.prod, Finset.range, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
13 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_6 : a 6 = 2 := by
…
```
Mentions: OeisA78729.a
theorem OeisA78729.conjecture $(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime sorry line 101 source
Source documentation
$(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime. Hence $a(4) = 0$. Conjecture: $a(n) = 0$ if and only if $n = 4$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Eq, OeisA78729.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 0 < n) : a n = 0 ↔ n = 4 := by
…
```
Mentions: OeisA78729.a