108301.lean`

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

8 declarations, 2 contain sorry, 0 declared as axioms

Digital sum of the Fermat number $2^{2^n} + 1$

a n is the digital sum of the Fermat number $2^{2^n} + 1$. The conjecture asks if there are any prime numbers in this sequence beyond $n=11$.

References:

A108301

Imports (1)

FormalConjecturesUtil

def OeisA108301.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. a n is the digital sum of the Fermat number $2^{2^n} + 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.sum, Nat.digits, OfNat.ofNat, HAdd.hAdd, HPow.hPow
9 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA108301.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 42 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108301.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_0 : a 0 = 3 := by decide
…
```
Mentions: OeisA108301.a
theorem OeisA108301.a_1 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108301.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 OeisA108301.a_2 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108301.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 OeisA108301.a_3 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108301.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 OeisA108301.a_4 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108301.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 OeisA108301.primes_in_a $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes sorry line 58 source
Source documentation
$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes.

depends on sorry what this means
Statement uses: And, Nat.Prime, OeisA108301.a, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem primes_in_a :
    (a 0).Prime ∧ (a 1).Prime ∧ (a 5).Prime ∧
    (a 6).Prime ∧ (a 7).Prime ∧ (a 11).Prime := by
…
```
Mentions: OeisA108301.a
theorem OeisA108301.conjecture $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes sorry line 65 source
Source documentation
$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes. Are there any more?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, Nat.Prime, OeisA108301.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 : answer(sorry) ↔ ∃ n > 11, (a n).Prime := by
…
```
Mentions: OeisA108301.a