78680.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime

The sequence $a(n)$ is the smallest positive integer $m$ such that $n \cdot 2^m + 1$ is prime, or $0$ if no such $m$ exists.

References:

A078680

Imports (1)

FormalConjecturesUtil

def OeisA78680.a Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime, or 0 if no such $m$ exists line 35 source
Source documentation
Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime, or 0 if no such $m$ 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, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not
13 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA78680.a_1 Value of the sequence a at 1 line 43 source
Source documentation
Value of the sequence a at 1.

extra axioms: _private.0.OeisA78680.a_1._native.decide.ax_1, _private.0.OeisA78680.a_1._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78680.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
18 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: OeisA78680.a
theorem OeisA78680.a_2 Value of the sequence a at 2 line 53 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA78680.a_2._native.decide.ax_1, _private.0.OeisA78680.a_2._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78680.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
18 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: OeisA78680.a
theorem OeisA78680.a_3 Value of the sequence a at 3 line 63 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA78680.a_3._native.decide.ax_1, _private.0.OeisA78680.a_3._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78680.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
18 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 = 1 := by
…
```
Mentions: OeisA78680.a
theorem OeisA78680.a_4 Value of the sequence a at 4 line 73 source
Source documentation
Value of the sequence a at 4.

extra axioms: _private.0.OeisA78680.a_4._native.decide.ax_1, _private.0.OeisA78680.a_4._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78680.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 2 := by
…
```
Mentions: OeisA78680.a
theorem OeisA78680.a_5 Value of the sequence a at 5 line 83 source
Source documentation
Value of the sequence a at 5.

extra axioms: _private.0.OeisA78680.a_5._native.decide.ax_1, _private.0.OeisA78680.a_5._native.decide.ax_2 what this means
Statement uses: Eq, Nat, OeisA78680.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul, HPow.hPow, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, False and 7 more
18 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 = 1 := by
…
```
Mentions: OeisA78680.a
theorem OeisA78680.conjecture There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to the statement that $2^{2^k} + 1$ is composite for $k > 4$) sorry line 96 source
Source documentation
There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to the statement that $2^{2^k} + 1$ is composite for $k > 4$). - T. D. Noe, Feb 25 2011

depends on sorry what this means
Statement uses: And, Eq, Nat, OeisA78680.a, HPow.hPow, OfNat.ofNat, LE.le, LT.lt, Ne
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    a (2 ^ 16) = 0 ∧ ∀ n : ℕ, 1 ≤ n ∧ n < 2 ^ 16 → a n ≠ 0 := by
…
```
Mentions: OeisA78680.a