67720.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Conjectures associated with A067720

A067720 lists numbers $k$ such that $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$, where $\varphi$ is Euler's totient function.

The sequence exhibits a strong connection to primes: for almost all terms $k$, $k + 1$ is prime. The conjecture states that $k = 8$ is the only exception.

References:

A067720

Imports (1)

FormalConjecturesUtil

Imported by 2 files

def OeisA67720.A A number $k$ is in the sequence A067720 if $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$ line 40 source
Source documentation
A number $k$ is in the sequence A067720 if $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Eq, Nat.totient, HAdd.hAdd, HPow.hPow, OfNat.ofNat, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
def A (k : ℕ) : Prop :=
…
```
theorem OeisA67720.a_1 $1$ is in the sequence A067720 line 45 source
Source documentation
$1$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, Eq.trans, True, congr, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.one_natPow, Nat.totient_two, mul_one, eq_self, eq_true and 1 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : A 1 := by norm_num [A]
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_2 $2$ is in the sequence A067720 line 49 source
Source documentation
$2$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, eq_true_of_decide, Eq.refl, Bool, Bool.true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : A 2 := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_4 $4$ is in the sequence A067720 line 54 source
Source documentation
$4$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, eq_true_of_decide, Eq.refl, Bool, Bool.true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : A 4 := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_6 $6$ is in the sequence A067720 line 59 source
Source documentation
$6$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, eq_true_of_decide, Eq.refl, Bool, Bool.true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_6 : A 6 := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_8 $8$ is in the sequence A067720 line 64 source
Source documentation
$8$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, eq_true_of_decide, Eq.refl, Bool, Bool.true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_8 : A 8 := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_10 $10$ is in the sequence A067720 line 69 source
Source documentation
$10$ is in the sequence A067720.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA67720.A, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq, Nat.totient, HAdd.hAdd, HPow.hPow, HMul.hMul, eq_true_of_decide, Eq.refl, Bool, Bool.true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_10 : A 10 := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.a_of_primes If $k + 1$ and $k^2 + 1$ are both prime, then $k$ is in the sequence line 74 source
Source documentation
If $k + 1$ and $k^2 + 1$ are both prime, then $k$ is in the sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, HAdd.hAdd, OfNat.ofNat, HPow.hPow, OeisA67720.A
Proof uses: Eq.mpr, Eq, Nat.totient, HMul.hMul, id, congrArg, OeisA67720.A.eq_1, HSub.hSub, Nat.totient_prime, Nat.add_sub_cancel, sq, Eq.refl
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_of_primes {k : ℕ} (hk : (k + 1).Prime) (hk' : (k ^ 2 + 1).Prime) : A k := by
…
```
Mentions: OeisA67720.A
theorem OeisA67720.prime_add_one_of_a For members of the sequence other than $8$, we have $k + 1$ is prime sorry line 79 source
Source documentation
For members of the sequence other than $8$, we have $k + 1$ is prime.

depends on sorry what this means
Statement uses: Nat, OeisA67720.A, Ne, OfNat.ofNat, Nat.Prime, HAdd.hAdd
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 prime_add_one_of_a {k : ℕ} (h : A k) (hne : k ≠ 8) : (k + 1).Prime := by
…
```
Mentions: OeisA67720.A