CarmichaelTotient.lean`

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5 declarations, 2 contain sorry, 0 declared as axioms

Carmichael's totient function conjecture

For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$ where $φ$ is the Euler totient function.

References:

Wikipedia
[F1998] Kevin Ford. The distribution of totients. https://arxiv.org/abs/1104.3264

Imports (1)

FormalConjecturesUtil

def CarmichaelTotient.CarmichaelTotientFor Natural number $n$ for which there exists a $m ≠ n$ with $φ(m) = φ(n)$ line 40 source
Source documentation
Natural number $n$ for which there exists a $m ≠ n$ with $φ(m) = φ(n)$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, And, Ne, Eq, Nat.totient
```
def CarmichaelTotientFor (n : ℕ) : Prop := ∃ m : ℕ, m ≠ n ∧ φ m = φ n
…
```
theorem CarmichaelTotient.carchimichealTotientFor_zero $n = 0 ↔ φ(n) = 0$ line 44 source
Source documentation
$n = 0 ↔ φ(n) = 0$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, CarmichaelTotient.CarmichaelTotientFor, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, False, True, congrArg, Exists, And, Ne, Eq, Nat.totient, funext, not_false_eq_true
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem carchimichealTotientFor_zero : ¬ CarmichaelTotientFor 0 := by
…
```
Mentions: CarmichaelTotient.CarmichaelTotientFor
theorem CarmichaelTotient.carmichealTotientFor_odd For every odd number $n$, $φ(2n) = φ(n)$ line 51 source
Source documentation
For every odd number $n$, $φ(2n) = φ(n)$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Odd, CarmichaelTotient.CarmichaelTotientFor
Proof uses: Exists.intro, And, Ne, Eq, Nat.totient, HMul.hMul, OfNat.ofNat, And.intro, Ne.symm, Nat.ne_of_lt, lt_two_mul_self, Odd.pos, Eq.mpr, id, congrArg, Nat.totient_mul, Iff.mpr, Nat.Coprime, Nat.coprime_two_left, Nat.totient_two and 2 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem carmichealTotientFor_odd {n : ℕ} (hn : Odd n) : CarmichaelTotientFor n := by
…
```
Mentions: CarmichaelTotient.CarmichaelTotientFor
theorem CarmichaelTotient.charmichaelTotient Carmichael's totient function conjecture*: For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$ sorry line 59 source
Source documentation
Carmichael's totient function conjecture: For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, CarmichaelTotient.CarmichaelTotientFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem charmichaelTotient :
    ∀ ⦃n : ℕ⦄, 0 < n → CarmichaelTotientFor n := by
…
```
Mentions: CarmichaelTotient.CarmichaelTotientFor
theorem CarmichaelTotient.carchimaelTotient_bound In Theorem 6 in [F1998], Kevin Ford proves that the smallest counterexample to Carmichael's totient function conjecture must be $≥ 10 ^ (10 ^ 10)$ sorry line 66 source
Source documentation
In Theorem 6 in [F1998], Kevin Ford proves that the smallest counterexample to Carmichael's totient function conjecture must be $≥ 10 ^ (10 ^ 10)$

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HPow.hPow, CarmichaelTotient.CarmichaelTotientFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem carchimaelTotient_bound {n : ℕ} (hn : 0 < n) (hn' : n < 10 ^ (10 ^ 10)) :
    CarmichaelTotientFor n := by
…
```
Mentions: CarmichaelTotient.CarmichaelTotientFor