RegularPrimes.lean`

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9 declarations, 5 contain sorry, 0 declared as axioms

Infinite Regular Primes

We define the notion of regular primes, which are prime numbers that are coprime with the cardinality of the class group of the p-th cyclotomic field. We also state that there are infinitely many regular primes.

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

def RegularPrimes.IsRegularPrime A natural prime number p is regular if p is coprime with the order of the class group of the p-th cyclotomic field line 40 source
Source documentation
A natural prime number p is regular if p is coprime with the order of the class group of the p-th cyclotomic field.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime
Proof uses: Nat.Coprime, Fintype.card, ClassGroup, NumberField.RingOfIntegers, CyclotomicField, Rat
7 instances, coercions and auxiliary constants left out.
```
noncomputable def IsRegularPrime [Fact p.Prime] : Prop :=
…
```
theorem RegularPrimes.not_isRegularPrime_37 The prime 37 is not a regular prime sorry line 45 source
Source documentation
The prime 37 is not a regular prime.

depends on sorry what this means
Statement uses: Not, RegularPrimes.IsRegularPrime, OfNat.ofNat, Nat
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: 2 steps
Open to load the checked before-and-after goal states.
```
theorem not_isRegularPrime_37 : ¬ @IsRegularPrime 37 (by decide) := by
…
```
Mentions: RegularPrimes.IsRegularPrime
def RegularPrimes.regularPrimes The set of regular primes line 49 source
Source documentation
The set of regular primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Nat.Prime, RegularPrimes.IsRegularPrime
1 instances, coercions and auxiliary constants left out.
```
def regularPrimes : Set ℕ := { p | ∃ (hp : Nat.Prime p), @IsRegularPrime p ⟨hp⟩ }
```
Mentions: RegularPrimes.IsRegularPrime
def RegularPrimes.irregularPrimes The set of irregular primes line 52 source
Source documentation
The set of irregular primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Nat.Prime, Not, RegularPrimes.IsRegularPrime
1 instances, coercions and auxiliary constants left out.
```
def irregularPrimes : Set ℕ := { p | ∃ (hp : Nat.Prime p), ¬ @IsRegularPrime p ⟨hp⟩ }
…
```
Mentions: RegularPrimes.IsRegularPrime
lemma RegularPrimes.small_regular_primes The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular sorry line 56 source
Source documentation
The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular.

depends on sorry what this means
Statement uses: LE.le, Set, Nat, Insert.insert, OfNat.ofNat, Singleton.singleton, RegularPrimes.regularPrimes
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma small_regular_primes :
    { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 } ⊆ regularPrimes := by
…
```
Mentions: RegularPrimes.regularPrimes
theorem RegularPrimes.isRegularPrime_iff_Bernoulli Kummer's criterion sorry line 64 source
Source documentation
Kummer's criterion. An equivalent definition of a regular prime p is that it does not divide the numerator of any of the Bernoulli numbers $B_k$ for $k = 2, 4, 6, \dots, p - 3$. Not in Mathlib.

depends on sorry what this means
Statement uses: Nat, Fact, Nat.Prime, Iff, RegularPrimes.IsRegularPrime, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, HSub.hSub, Even, Not, Dvd.dvd, Int, Rat.num, bernoulli'
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isRegularPrime_iff_Bernoulli (p : ℕ) [Fact p.Prime] :
    IsRegularPrime p ↔ ∀ k ∈ Finset.Icc 2 (p - 3), Even k → ¬ (p : ℤ) ∣ (bernoulli' k).num := by
…
```
Mentions: RegularPrimes.IsRegularPrime
theorem RegularPrimes.infinitude_of_irregularprimes The set of irregular primes is infinite sorry line 70 source
Source documentation
The set of irregular primes is infinite.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, RegularPrimes.irregularPrimes
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 infinitude_of_irregularprimes : irregularPrimes.Infinite := by
…
```
def RegularPrimes.RegularPrimeConjecture Conjecture: The set of regular primes is infinite line 74 source
Source documentation
Conjecture: The set of regular primes is infinite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Set.Infinite, Nat, RegularPrimes.regularPrimes
```
def RegularPrimeConjecture : Prop :=
…
```
theorem RegularPrimes.regularprime_conjecture Conjecture: The set of regular primes is infinite sorry line 79 source
Source documentation
Conjecture: The set of regular primes is infinite.

depends on sorry what this means
Statement uses: RegularPrimes.RegularPrimeConjecture
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 regularprime_conjecture : RegularPrimeConjecture := by
…
```
Mentions: RegularPrimes.RegularPrimeConjecture