113609.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Number of prime powers $q<=n$ such that also $q+2$ is a prime power

References:

A113609

Imports (1)

FormalConjecturesUtil

def OeisA113609.IsOeisPrimePower A number $n$ is an "OEIS prime power" (for the context of A113609's definition) if $n=1$ or $n$ is a standard prime power line 35 source
Source documentation
A number $n$ is an "OEIS prime power" (for the context of A113609's definition) if $n=1$ or $n$ is a standard prime power.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Or, Eq, OfNat.ofNat, IsPrimePow
2 instances, coercions and auxiliary constants left out.
```
def IsOeisPrimePower (n : ℕ) : Prop := n = 1 ∨ IsPrimePow n
```
instance OeisA113609.DecidableIsOeisPrimePower line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Decidable, OeisA113609.IsOeisPrimePower
Proof uses: id, Eq, OfNat.ofNat, IsPrimePow
5 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def OeisA113609.a The primary defining sequence a line 46 source
Source documentation
The primary defining sequence a. $a(n)$ is the number of prime powers $q<=n$ such that also $q+2$ is a prime power. $$a(n) = \operatorname{card} \{q \in \mathbb{N} \mid 1 \le q \le n \land P(q) \land P(q+2) \}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, And, OeisA113609.IsOeisPrimePower, HAdd.hAdd, OfNat.ofNat, GE.ge, Finset.range
7 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA113609.a_1 line 51 source
extra axioms: _private.0.OeisA113609.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA113609.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA113609.a_2 line 54 source
extra axioms: _private.0.OeisA113609.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA113609.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA113609.a_3 line 57 source
extra axioms: _private.0.OeisA113609.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA113609.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA113609.a_4 line 60 source
extra axioms: _private.0.OeisA113609.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA113609.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA113609.a_5 line 63 source
extra axioms: _private.0.OeisA113609.a_5._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA113609.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA113609.conjecture (25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6) sorry line 70 source
Source documentation
(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GE.ge, OfNat.ofNat, OeisA113609.IsOeisPrimePower, HAdd.hAdd, Not, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 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) ↔ ∃ q ≥ 1000000,
    IsOeisPrimePower q ∧ IsOeisPrimePower (q + 2) ∧
    ¬ q.Prime ∧ ¬ (q + 2).Prime := by
…
```
Mentions: OeisA113609.IsOeisPrimePower