81091.lean`

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8 declarations, 1 contain sorry, 0 declared as axioms

Primes of the form 2^n + 2^i + 1

There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$. See Wagstaff (2001) where this conjecture is posed.

References:

A81091
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267.

Imports (1)

FormalConjecturesUtil

def OeisA81091.isPrimeBitsSet Primes with $m$ one bits in their binary representation line 37 source
Source documentation
Primes with $m$ one bits in their binary representation.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Nat.Prime, Eq, List.count, Bool, Bool.true, Nat.bits
2 instances, coercions and auxiliary constants left out.
```
def isPrimeBitsSet (m p : ℕ) : Prop :=
…
```
def OeisA81091.A Primes in A81091 have exactly $3$ set bits in binary representation ($2^n + 2^i + 1$) line 41 source
Source documentation
Primes in A81091 have exactly $3$ set bits in binary representation ($2^n + 2^i + 1$).

standard axioms: Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA81091.isPrimeBitsSet, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def A (p : ℕ) : Prop :=
…
```
theorem OeisA81091.a_7 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA81091.A, OfNat.ofNat, Nat
Proof uses: id, OeisA81091.isPrimeBitsSet
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA81091.a_11 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA81091.A, OfNat.ofNat, Nat
Proof uses: id, OeisA81091.isPrimeBitsSet
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA81091.a_13 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA81091.A, OfNat.ofNat, Nat
Proof uses: id, OeisA81091.isPrimeBitsSet
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA81091.a_19 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA81091.A, OfNat.ofNat, Nat
Proof uses: id, OeisA81091.isPrimeBitsSet
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA81091.a_37 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA81091.A, OfNat.ofNat, Nat
Proof uses: id, OeisA81091.isPrimeBitsSet
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA81091.conjectureA81091 Conjecture (A81091): There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$ sorry line 64 source
Source documentation
Conjecture (A81091): There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$.

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, OeisA81091.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjectureA81091 :
    answer(sorry) ↔ Set.Infinite {p : ℕ | A p} := by
…
```
Mentions: OeisA81091.A