WilsonPrime.lean`

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6 declarations, 1 contain sorry, 0 declared as axioms

Wilson primes

A Wilson prime is a prime $p$ for which $p^2$ divides $(p-1)!+1$. The only known examples are $5$, $13$, and $563$. It is conjectured that infinitely many Wilson primes exist.

References:

Wikipedia, Wilson prime
OEIS A007540
E. Costa, R. Gerbicz, and D. Harvey, A search for Wilson primes

Imports (1)

FormalConjecturesUtil

def WilsonPrime.IsWilsonPrime A Wilson prime is a prime $p$ such that $p^2 \mid (p-1)!+1$ line 38 source
Source documentation
A Wilson prime is a prime $p$ such that $p^2 \mid (p-1)!+1$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Nat.Prime, Dvd.dvd, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Nat.factorial, HSub.hSub
10 instances, coercions and auxiliary constants left out.
```
def IsWilsonPrime (p : ℕ) : Prop :=
…
```
theorem WilsonPrime.infinitely_many_wilson_primes There are infinitely many Wilson primes sorry line 43 source
Source documentation
There are infinitely many Wilson primes.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, WilsonPrime.IsWilsonPrime
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 infinitely_many_wilson_primes : Set.Infinite {p : ℕ | IsWilsonPrime p} := by
…
```
Mentions: WilsonPrime.IsWilsonPrime
theorem WilsonPrime.isWilsonPrime_five The prime $5$ is a Wilson prime line 48 source
Source documentation
The prime $5$ is a Wilson prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WilsonPrime.IsWilsonPrime, OfNat.ofNat, Nat
Proof uses: of_eq_true, And, Nat.Prime, Dvd.dvd, HPow.hPow, HAdd.hAdd, Nat.factorial, HSub.hSub, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl and 11 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isWilsonPrime_five : IsWilsonPrime 5 := by
…
```
Mentions: WilsonPrime.IsWilsonPrime
theorem WilsonPrime.isWilsonPrime_thirteen The prime $13$ is a Wilson prime line 53 source
Source documentation
The prime $13$ is a Wilson prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WilsonPrime.IsWilsonPrime, OfNat.ofNat, Nat
Proof uses: of_eq_true, And, Nat.Prime, Dvd.dvd, HPow.hPow, HAdd.hAdd, Nat.factorial, HSub.hSub, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl and 12 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isWilsonPrime_thirteen : IsWilsonPrime 13 := by
…
```
Mentions: WilsonPrime.IsWilsonPrime
theorem WilsonPrime.not_isWilsonPrime_one The primality condition excludes $1$, which satisfies the divisibility condition alone line 58 source
Source documentation
The primality condition excludes $1$, which satisfies the divisibility condition alone.

standard axioms: propext what this means
Statement uses: Not, WilsonPrime.IsWilsonPrime, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, True, False, congrArg, And, Nat.Prime, Dvd.dvd, HPow.hPow, HAdd.hAdd, Nat.factorial, HSub.hSub, congr, eq_false, Mathlib.Meta.NormNum.isNat_prime_1, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, eq_true, Mathlib.Meta.NormNum.isNat_dvd_true, Mathlib.Meta.NormNum.isNat_pow and 8 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem not_isWilsonPrime_one : ¬ IsWilsonPrime 1 := by
…
```
Mentions: WilsonPrime.IsWilsonPrime
theorem WilsonPrime.not_isWilsonPrime_seven The prime $7$ is not a Wilson prime line 63 source
Source documentation
The prime $7$ is not a Wilson prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, WilsonPrime.IsWilsonPrime, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, True, False, congrArg, And, Nat.Prime, Dvd.dvd, HPow.hPow, HAdd.hAdd, Nat.factorial, HSub.hSub, congr, eq_true, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 16 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem not_isWilsonPrime_seven : ¬ IsWilsonPrime 7 := by
…
```
Mentions: WilsonPrime.IsWilsonPrime