224.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

7 declarations, 1 contain sorry, 0 declared as axioms

Number of squares $\bmod n$

The number of squares modulo $n$. This is the cardinality of the set $\{k^2 \bmod n \mid k \in \{0, 1, \dots, n-1\}\}$.

References:

A000224

Imports (1)

FormalConjecturesUtil

def OeisA224.a The number of squares modulo $n$ line 35 source
Source documentation
The number of squares modulo $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, Finset.card, Finset.image, HMod.hMod, HPow.hPow, Finset.range
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA224.a_0 Value of the sequence a at 0 line 42 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by decide
…
```
Mentions: OeisA224.a
theorem OeisA224.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 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 = 1 := by decide
…
```
Mentions: OeisA224.a
theorem OeisA224.a_2 Value of the sequence a at 2 line 50 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 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 = 2 := by decide
…
```
Mentions: OeisA224.a
theorem OeisA224.a_3 Value of the sequence a at 3 line 54 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2 := by decide
…
```
Mentions: OeisA224.a
theorem OeisA224.a_4 Value of the sequence a at 4 line 58 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA224.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 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 = 2 := by decide
…
```
Mentions: OeisA224.a
theorem OeisA224.conjecture $n^2 \equiv 1 \pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime sorry line 65 source
Source documentation
$n^2 \equiv 1 \pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime.
- Thomas Ordowski, Jun 08 2017
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, And, Nat.Prime, Ne, Nat.ModEq, HMul.hMul, OeisA224.a, HSub.hSub, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 1 < n) :
    (n.Prime ∧ n ≠ 2) ↔ n ^ 2 ≡ 1 [MOD a n * (a n - 1)] := by
…
```
Mentions: OeisA224.a