409.lean`

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13 declarations, 12 contain sorry, 0 declared as axioms

Erdős Problem 409

Reference: erdosproblems.com/409

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Erdos409.erdos_409.parts.i How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached? sorry line 40 source
Source documentation
How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached?

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HAdd.hAdd, Nat.totient, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.i (n : ℕ) (hn : 0 < n) :
    IsLeast { i | (φ · + 1)^[i] n |>.Prime } answer(sorry) := by
…
```
theorem Erdos409.erdos_409.variants.termination If $n > 0$, then the iteration $n\mapsto\phi(n) + 1$ necessarily reaches a prime line 47 source
Source documentation
If $n > 0$, then the iteration $n\mapsto\phi(n) + 1$ necessarily reaches a prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, Nat.Prime, Nat.iterate, HAdd.hAdd, Nat.totient
Proof uses: Nat.strong_induction_on, dite, Exists.intro, id, Not, Or, Eq, Decidable.byContradiction, of_eq_true, Eq.trans, True, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Bool, Bool.true and 16 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.variants.termination (n : ℕ) (hn : 0 < n) :
    ∃ i, (φ · + 1)^[i] n |>.Prime := by
…
```
theorem Erdos409.erdos_409.parts.i.isTheta Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached sorry line 80 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. What is $\Theta(c(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HAdd.hAdd, Nat.totient, Asymptotics.IsTheta, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.i.isTheta (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c, Erdos409.erdos_409.parts.i
theorem Erdos409.erdos_409.parts.i.isBigO Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached sorry line 90 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HAdd.hAdd, Nat.totient, Asymptotics.IsBigO, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.i.isBigO (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c, Erdos409.erdos_409.parts.i
theorem Erdos409.erdos_409.parts.i.isLittleO Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached sorry line 100 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HAdd.hAdd, Nat.totient, Asymptotics.IsLittleO, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.i.isLittleO (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c, Erdos409.erdos_409.parts.i
theorem Erdos409.erdos_409.parts.ii Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$? sorry line 109 source
Source documentation
Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Nat.Prime, Set.Infinite, Set.ofPred, Eq, Nat.iterate, HAdd.hAdd, Nat.totient, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.ii :
    answer(sorry) ↔ ∃ (p : ℕ) (hp : p.Prime), { n | ∃ i, (φ · + 1)^[i] n = p }.Infinite := by
…
```
Mentions: Erdos409.erdos_409.parts.i
theorem Erdos409.erdos_409.parts.iii What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$? sorry line 117 source
Source documentation
What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$?

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Set.HasDensity, Set.ofPred, Exists, Eq, Nat.iterate, HAdd.hAdd, Nat.totient, OfNat.ofNat, sorryAx, Real, Bool.false, Set.univ
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.parts.iii (p : ℕ) (h : p.Prime) :
    { n | ∃ i, (φ · + 1)^[i] n = p }.HasDensity answer(sorry) := by
…
```
Mentions: Erdos409.erdos_409.parts.i
theorem Erdos409.erdos_409.variants.sigma If the iteration $n\mapsto\sigma(n) - 1$ starting at $n > 1$ reaches a prime, how many iterations are needed before a prime is reached? sorry line 128 source
Source documentation
If the iteration $n\mapsto\sigma(n) - 1$ starting at $n > 1$ reaches a prime, how many iterations are needed before a prime is reached?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Exists, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma, IsLeast, Set.ofPred, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.variants.sigma (n : ℕ) (hn : n > 1)
    (hterm : ∃ i, (σ 1 · - 1)^[i] n |>.Prime) :
    IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } answer(sorry) := by
…
```
theorem Erdos409.erdos_409.variants.sigma_termination If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime sorry line 137 source
Source documentation
If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime. Note: this is open — it is not clear that the σ iteration always terminates, since it is non-decreasing (unlike the φ iteration which is strictly decreasing).

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Exists, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma
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 erdos_409.variants.sigma_termination (n : ℕ) (hn : n > 1) :
    ∃ i, (σ 1 · - 1)^[i] n |>.Prime := by
…
```
theorem Erdos409.erdos_409.variants.sigma_isTheta Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached sorry line 148 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma, Asymptotics.IsTheta, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.variants.sigma_isTheta (c : ℕ → ℕ)
    (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c
theorem Erdos409.erdos_409.variants.sigma_isBigO Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached sorry line 158 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma, Asymptotics.IsBigO, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.variants.sigma_isBigO (c : ℕ → ℕ)
    (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c
theorem Erdos409.erdos_409.variants.sigma_isLittleO Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached sorry line 168 source
Source documentation
Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, IsLeast, Set.ofPred, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma, Asymptotics.IsLittleO, Real, Filter.atTop, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_409.variants.sigma_isLittleO (c : ℕ → ℕ)
    (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: c
theorem Erdos409.erdos_409.variants.sigma_prime_termination Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime? sorry line 177 source
Source documentation
Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime?

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Exists, Nat.Prime, Nat.iterate, HSub.hSub, ArithmeticFunction, ArithmeticFunction.sigma
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 erdos_409.variants.sigma_prime_termination :
    answer(sorry) ↔ ∀ n > 1, ∃ i, (σ 1 · - 1)^[i] n |>.Prime := by
…
```