64.lean`

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5 declarations, 2 contain sorry, 0 declared as axioms

Green's Open Problem 64

Reference: Ben Green's Open Problems

Do there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors, counted with multiplicity?

Imports (1)

FormalConjecturesUtil

theorem Green64.green_64 Do there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors, counted with multiplicity (i.e. $\Omega(p - 2)$ is odd)? sorry line 42 source
Source documentation
Do there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors, counted with multiplicity (i.e. $\Omega(p - 2)$ is odd)?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, Nat.Prime, Odd, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_64 :
    answer(sorry) ↔ {p : ℕ | p.Prime ∧ Odd (Ω (p - 2))}.Infinite := by
…
```
theorem Green64.green_64_mem_five $5$ satisfies the condition: $5$ is prime and $5 - 2 = 3$ is prime, so $\Omega(3) = 1$ is odd line 48 source
Source documentation
$5$ satisfies the condition: $5$ is prime and $5 - 2 = 3$ is prime, so $\Omega(3) = 1$ is odd.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Nat, Set, Set.ofPred, And, Nat.Prime, Odd, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub, OfNat.ofNat
Proof uses: And.intro, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_0, Bool.false, Eq.mpr, id, Eq, congrArg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natSub, ArithmeticFunction.cardFactors_apply_prime, odd_one
18 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem green_64_mem_five : 5 ∈ {p : ℕ | p.Prime ∧ Odd (Ω (p - 2))} := by
…
```
theorem Green64.green_64_mem_seven $7$ satisfies the condition: $7$ is prime and $7 - 2 = 5$ is prime, so $\Omega(5) = 1$ is odd line 55 source
Source documentation
$7$ satisfies the condition: $7$ is prime and $7 - 2 = 5$ is prime, so $\Omega(5) = 1$ is odd.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Nat, Set, Set.ofPred, And, Nat.Prime, Odd, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub, OfNat.ofNat
Proof uses: And.intro, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_0, Bool.false, Eq.mpr, id, Eq, congrArg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natSub, ArithmeticFunction.cardFactors_apply_prime, odd_one
18 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem green_64_mem_seven : 7 ∈ {p : ℕ | p.Prime ∧ Odd (Ω (p - 2))} := by
…
```
theorem Green64.green_64_not_mem_eleven $11$ does not satisfy the condition: although $11$ is prime, $11 - 2 = 9 = 3 ^ 2$ has $\Omega(9) = 2$ prime factors, which is even line 63 source
Source documentation
$11$ does not satisfy the condition: although $11$ is prime, $11 - 2 = 9 = 3 ^ 2$ has $\Omega(9) = 2$ prime factors, which is even. This shows the condition is non-trivial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Membership.mem, Nat, Set, Set.ofPred, And, Nat.Prime, Odd, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub, OfNat.ofNat
Proof uses: False, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mp, HPow.hPow, congrArg, ArithmeticFunction.cardFactors_apply_prime_pow, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_0, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natSub and 3 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem green_64_not_mem_eleven : 11 ∉ {p : ℕ | p.Prime ∧ Odd (Ω (p - 2))} := by
…
```
theorem Green64.green_64.variants.p_sub_one The same question as green_64 but with $p - 1$ instead of $p - 2$ sorry line 73 source
Source documentation
The same question as green_64 but with $p - 1$ instead of $p - 2$. Green notes this is "probably more natural".

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, Nat.Prime, Odd, ArithmeticFunction, ArithmeticFunction.cardFactors, HSub.hSub, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_64.variants.p_sub_one :
    answer(sorry) ↔ {p : ℕ | p.Prime ∧ Odd (Ω (p - 1))}.Infinite := by
…
```