113213.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 number $m$ such that $2^n - m$ and $2^n + m$ are primes

References:

A113213

Imports (1)

FormalConjecturesUtil

def OeisA113213.a a n is the smallest number m such that $2^n - m$ and $2^n + m$ are primes line 39 source
Source documentation
a n is the smallest number m such that $2^n - m$ and $2^n + m$ are primes. Computed by checking elements from $0$ to $2^n$, as any larger $m$ would make $2^n - m = 0$ which is not prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List, List.filter, Decidable.decide, And, Nat.Prime, HSub.hSub, HPow.hPow, OfNat.ofNat, HAdd.hAdd, List.range, List.headD
11 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA113213.a_1 line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113213.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 OeisA113213.a_2 line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113213.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 OeisA113213.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113213.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 OeisA113213.a_4 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113213.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 OeisA113213.a_5 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113213.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 OeisA113213.conjecture Conjecture: $a(n) = O(n^3)$ sorry line 66 source
Source documentation
Conjecture: $a(n) = O(n^3)$.
The source defines $a(n)$ as the least $m$ with $2^n - m$ and $2^n + m$ prime, so it implicitly asserts that such an $m$ exists. Since a n = 0 when no such $m$ exists, the existence of a prime pair is stated explicitly for all sufficiently large $n$.

depends on sorry what this means
Statement uses: And, Filter.Eventually, Nat, Exists, Nat.Prime, HSub.hSub, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Filter.atTop, Asymptotics.IsBigO, Real, OeisA113213.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    (∀ᶠ n : ℕ in Filter.atTop, ∃ m, (2 ^ n - m).Prime ∧ (2 ^ n + m).Prime) ∧
    (fun n : ℕ => (a n : ℝ)) =O[Filter.atTop] (fun n : ℕ => (n ^ 3 : ℝ)) := by
…
```
Mentions: OeisA113213.a