AgohGiuga.lean`

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

19 declarations, 12 contain sorry, 0 declared as axioms

Agoh-Giuga conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def AgohGiuga.AgohGiugaCongr The Agoh-Giuga Conjecture, Agoh's formulation line 50 source
Source documentation
The Agoh-Giuga Conjecture, Agoh's formulation. An integer p ≥ 2 is prime if and only if we have p*B_{p-1} ≡ -1 [MOD p]

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Nat, GE.ge, OfNat.ofNat, Iff, Nat.Prime, Exists, Int, Rat, bernoulli', HSub.hSub, Eq, HAdd.hAdd, HMul.hMul, Rat.num, Rat.den, HPow.hPow
14 instances, coercions and auxiliary constants left out.
```
def AgohGiugaCongr : Prop :=
…
```
def AgohGiuga.AgohGiugaSum The Agoh-Giuga Conjecture, Giuga's formulation line 60 source
Source documentation
The Agoh-Giuga Conjecture, Giuga's formulation. An integer p ≥ 2 is prime if and only if it satisfies the congruence ∑_{i=1}^{p-1} i^{p-1} ≡ -1 [MOD p].

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Nat, GE.ge, OfNat.ofNat, Iff, Nat.Prime, Dvd.dvd, HAdd.hAdd, Finset.sum, Finset.Ioo, HPow.hPow, HSub.hSub
14 instances, coercions and auxiliary constants left out.
```
def AgohGiugaSum : Prop := ∀ p ≥ 2, p.Prime ↔
…
```
theorem AgohGiuga.agoh_giuga The Agoh-Giuga Conjecture, Agoh's formulation sorry line 65 source
Source documentation
The Agoh-Giuga Conjecture, Agoh's formulation

depends on sorry what this means
Statement uses: AgohGiuga.AgohGiugaCongr
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem agoh_giuga : AgohGiugaCongr := by
…
```
Mentions: AgohGiuga.AgohGiugaCongr
theorem AgohGiuga.agoh_giuga.variants.giuga The Agoh-Giuga Conjecture, Giuga's formulation sorry line 70 source
Source documentation
The Agoh-Giuga Conjecture, Giuga's formulation

depends on sorry what this means
Statement uses: AgohGiuga.AgohGiugaSum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem agoh_giuga.variants.giuga : AgohGiugaSum := by
…
```
Mentions: AgohGiuga.AgohGiugaSum
theorem AgohGiuga.agoh_giuga.variants.equivalence The two statements of the conjecture are equivalent sorry line 75 source
Source documentation
The two statements of the conjecture are equivalent.

depends on sorry what this means
Statement uses: Iff, AgohGiuga.AgohGiugaCongr, AgohGiuga.AgohGiugaSum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem agoh_giuga.variants.equivalence : AgohGiugaCongr ↔ AgohGiugaSum := by
…
```
Mentions: AgohGiuga.AgohGiugaCongr, AgohGiuga.AgohGiugaSum
def AgohGiuga.IsWeakGiuga A (weak) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{\varphi(n)} \equiv -1\pmod{n}$$ line 84 source
Source documentation
A (weak) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{\varphi(n)} \equiv -1\pmod{n}$$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Nat.Composite, Dvd.dvd, HAdd.hAdd, OfNat.ofNat, Finset.sum, Finset.Ioo, HPow.hPow, Nat.totient
11 instances, coercions and auxiliary constants left out.
```
def IsWeakGiuga (n : ℕ) : Prop :=
…
```
def AgohGiuga.IsStrongGiuga A (strong) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{n - 1} \equiv -1\pmod{n}$$ line 92 source
Source documentation
A (strong) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{n - 1} \equiv -1\pmod{n}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Nat.Composite, Dvd.dvd, HAdd.hAdd, OfNat.ofNat, Finset.sum, Finset.Ioo, HPow.hPow, HSub.hSub
13 instances, coercions and auxiliary constants left out.
```
def IsStrongGiuga (n : ℕ) : Prop :=
…
```
theorem AgohGiuga.isWeakGiuga_iff_prime_dvd sorry line 101 source
depends on sorry what this means
Statement uses: Nat, Nat.Composite, Iff, AgohGiuga.IsWeakGiuga, Membership.mem, Finset, Nat.primeFactors, Dvd.dvd, HSub.hSub, HDiv.hDiv, 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 AgohGiuga.isWeakGiuga_iff_sum_primeFactors A composite number $n$ is weak Giuga if and only if $$ \sum_{p\mid n} \frac{1}{p} - \frac{1}{n} \in\mathbb{N}. $$ sorry line 112 source
Source documentation
A composite number $n$ is weak Giuga if and only if $$ \sum_{p\mid n} \frac{1}{p} - \frac{1}{n} \in\mathbb{N}. $$

depends on sorry what this means
Statement uses: Nat, Nat.Composite, Iff, AgohGiuga.IsWeakGiuga, Exists, Eq, Rat, HSub.hSub, Finset.sum, Nat.primeFactors, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isWeakGiuga_iff_sum_primeFactors {n : ℕ} (hn : n.Composite) :
    IsWeakGiuga n ↔ ∃ m : ℕ, ∑ p ∈ n.primeFactors, (1 / p : ℚ) - 1 / n = m := by
…
```
Mentions: AgohGiuga.IsWeakGiuga
theorem AgohGiuga.squarefree_of_isCarmichael A composite Carmichael number is squarefree line 118 source
Source documentation
A composite Carmichael number is squarefree.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Composite, IsCarmichael, Squarefree
Proof uses: Eq.mpr, Nat.Prime, Not, Dvd.dvd, HMul.hMul, id, Eq, propext, Nat.squarefree_iff_prime_squarefree, False, And, LT.lt, OfNat.ofNat, LE.le, Nat.Coprime, HSub.hSub, HPow.hPow, absurd, HAdd.hAdd, Nat.le_add_left and 57 more
123 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem squarefree_of_isCarmichael {a : ℕ} (ha₁ : a.Composite) (ha₂ : IsCarmichael a) :
    Squarefree a := by
…
```
Mentions: IsCarmichael
theorem AgohGiuga.korselts_criterion A composite number a is Carmichael if and only if it is squarefree and, for all prime p dividing a, we have p - 1 ∣ a - 1 line 134 source
Source documentation
A composite number a is Carmichael if and only if it is squarefree and, for all prime p dividing a, we have p - 1 ∣ a - 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Composite, Iff, IsCarmichael, And, Squarefree, Nat.Prime, Dvd.dvd, HSub.hSub, OfNat.ofNat
Proof uses: Iff.intro, And.intro, AgohGiuga.squarefree_of_isCarmichael, Fact, Fact.mk, Exists, Units, ZMod, Membership.mem, Subgroup, Subgroup.zpowers, IsCyclic.exists_generator, Eq, HMul.hMul, Nat.Coprime, Decidable.byContradiction, Not, False, Iff.mp, Classical.not_not and 102 more
179 instances, coercions and auxiliary constants left out.

Tactic trace: 45 steps
Open to load the checked before-and-after goal states.
```
theorem korselts_criterion (a : ℕ) (ha₁ : a.Composite) :
    IsCarmichael a ↔ Squarefree a ∧
      ∀ p, p.Prime → p ∣ a → (p - 1 : ℕ) ∣ (a - 1 : ℕ) := by
…
```
Mentions: IsCarmichael
lemma AgohGiuga.isCarmichael_561 line 178 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsCarmichael, OfNat.ofNat, Nat
Proof uses: Nat.Composite, id, And.intro, LT.lt, Not, Nat.Prime, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_not_prime, Mathlib.Meta.NormNum.not_prime_mul_of_ble, Iff.mpr, And, Squarefree, Dvd.dvd, HSub.hSub, AgohGiuga.korselts_criterion, Eq and 30 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
theorem AgohGiuga.isStrongGiuga_iff Giuga showed that a number n is strong Giuga if and only if it is Carmichael and ∑_{p|n} 1/p - 1/n ∈ ℕ (i.e., if and only if it is Carmichael and weak Giuga) sorry line 206 source
Source documentation
Giuga showed that a number n is strong Giuga if and only if it is Carmichael and ∑_{p|n} 1/p - 1/n ∈ ℕ (i.e., if and only if it is Carmichael and weak Giuga). Ref: G. Giuga, Su una presumibile proprieta caratteristica dei numeri primi

depends on sorry what this means
Statement uses: Nat, Nat.Composite, Iff, AgohGiuga.IsStrongGiuga, And, IsCarmichael, Exists, Eq, Rat, HSub.hSub, Finset.sum, Nat.primeFactors, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isStrongGiuga_iff {a : ℕ} (ha : a.Composite) :
    IsStrongGiuga a ↔ IsCarmichael a ∧ ∃ n : ℕ, ∑ p ∈ a.primeFactors, (1 / p : ℚ) - 1 / a = n := by
…
```
Mentions: AgohGiuga.IsStrongGiuga, IsCarmichael
theorem AgohGiuga.agoh_giuga.variants.isStrongGiuga_implies_isCarmichael Every strong Giuga number is a Carmichael number sorry line 214 source
Source documentation
Every strong Giuga number is a Carmichael number.

depends on sorry what this means
Statement uses: Nat, AgohGiuga.IsStrongGiuga, IsCarmichael
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem agoh_giuga.variants.isStrongGiuga_implies_isCarmichael
    (a : ℕ) (ha : IsStrongGiuga a) : IsCarmichael a := by
…
```
Mentions: AgohGiuga.IsStrongGiuga, IsCarmichael
theorem AgohGiuga.agoh_giuga.variants.le_primeFactors_card_of_isStrongGiuga Giuga showed that a Giuga number has at least 9 prime factors sorry line 223 source
Source documentation
Giuga showed that a Giuga number has at least 9 prime factors. Ref: G. Giuga, Su una presumibile proprieta caratteristica dei numeri primi

depends on sorry what this means
Statement uses: Nat, AgohGiuga.IsStrongGiuga, LE.le, OfNat.ofNat, Finset.card, Nat.primeFactors
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 agoh_giuga.variants.le_primeFactors_card_of_isStrongGiuga
    (a : ℕ) (ha : IsStrongGiuga a) : 9 ≤ a.primeFactors.card := by
…
```
Mentions: AgohGiuga.IsStrongGiuga
theorem AgohGiuga.agoh_giuga.variants._1000_le_digits_length_of_isStrongGiuga Giuga showed that a counterexample Giuga number has at least 1000 digits sorry line 232 source
Source documentation
Giuga showed that a counterexample Giuga number has at least 1000 digits. Ref: G. Giuga, Su una presumibile proprieta caratteristica dei numeri primi

depends on sorry what this means
Statement uses: Nat, AgohGiuga.IsStrongGiuga, LE.le, OfNat.ofNat, List.length, Nat.digits
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 agoh_giuga.variants._1000_le_digits_length_of_isStrongGiuga
    (a : ℕ) (ha : IsStrongGiuga a) : 1000 ≤ (Nat.digits 10 a).length := by
…
```
Mentions: AgohGiuga.IsStrongGiuga
theorem AgohGiuga.agoh_giuga.variants._1700_le_digits_length_of_isStrongGiuga Bedocchi showed that any Giuga number has at least 1700 digits sorry line 241 source
Source documentation
Bedocchi showed that any Giuga number has at least 1700 digits. Ref: E. Bedocchi, Note on a conjecture about prime numbers

depends on sorry what this means
Statement uses: Nat, AgohGiuga.IsStrongGiuga, GT.gt, List.length, Nat.digits, OfNat.ofNat
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 agoh_giuga.variants._1700_le_digits_length_of_isStrongGiuga
    (a : ℕ) (ha : IsStrongGiuga a) :
    (Nat.digits 10 a).length > 1700 := by
…
```
Mentions: AgohGiuga.IsStrongGiuga
theorem AgohGiuga.agoh_giuga.variants._13000_le_digits_length_of_isStrongGiuga Borwein, Borwein, Borwein and Girgensohn showed that any strong Giuga number has at least 13000 digits sorry line 252 source
Source documentation
Borwein, Borwein, Borwein and Girgensohn showed that any strong Giuga number has at least 13000 digits. Ref: D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn, Giuga’s conjecture on primality

depends on sorry what this means
Statement uses: Nat, AgohGiuga.IsStrongGiuga, LE.le, OfNat.ofNat, List.length, Nat.digits
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 agoh_giuga.variants._13000_le_digits_length_of_isStrongGiuga
    (a : ℕ) (ha : IsStrongGiuga a) : 13000 ≤ (Nat.digits 10 a).length := by
…
```
Mentions: AgohGiuga.IsStrongGiuga
theorem AgohGiuga.agoh_giuga.variants.isStrongGiuga_growth Let G(X) denote the number of exceptions n ≤ X to Giuga’s conjecture sorry line 264 source
Source documentation
Let G(X) denote the number of exceptions n ≤ X to Giuga’s conjecture. Then for X larger than an absolute constant which can be made explicit, G(X) ≪ X^{1/2} log X. Ref: Vicentiu Tipu, A Note on Giuga’s Conjecture

depends on sorry what this means
Statement uses: Nat, Eq, Finset.card, Finset.filter, AgohGiuga.IsStrongGiuga, Classical.propDecidable, Finset.Icc, OfNat.ofNat, Exists, Real, GE.ge, LE.le, HMul.hMul, Real.sqrt, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, 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 agoh_giuga.variants.isStrongGiuga_growth
    (G : ℕ → ℕ) (hG : G = fun x => Finset.Icc 1 x |>.filter IsStrongGiuga |>.card) :
    ∃ N O, ∀ n ≥ N, G n ≤ O * (n : ℝ).sqrt * (n : ℝ).log := by
…
```
Mentions: AgohGiuga.IsStrongGiuga