WolstenholmePrime.lean`

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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, 4 contain sorry, 0 declared as axioms

Wolstenholme Prime

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

theorem WolstenholmePrime.wolstenholme_theorem sorry line 39 source
depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Nat.Prime, Nat.ModEq, HPow.hPow, Nat.choose, HSub.hSub, HMul.hMul
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.
def WolstenholmePrime.IsWolstenholmePrime A prime $p > 7$ is called a Wolstenholme prime if $\binom{2p-1}{p-1} \equiv 1 (\pmod{p^4})$ line 47 source
Source documentation
A prime $p > 7$ is called a Wolstenholme prime if $\binom{2p-1}{p-1} \equiv 1 (\pmod{p^4})$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, GT.gt, OfNat.ofNat, Nat.Prime, Nat.ModEq, HPow.hPow, Nat.choose, HSub.hSub, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
def IsWolstenholmePrime (p : ℕ) : Prop :=
…
```
theorem WolstenholmePrime.wolstenholme_prime_16483 Two known Wolstenholme primes: 16843 and 2124679 line 56 source
Source documentation
Two known Wolstenholme primes: 16843 and 2124679.
Formal proof linked here provided by AlphaProof

extra axioms: _private.0.WolstenholmePrime.wolstenholme_prime_16483._native.native_decide.ax_1 what this means
Statement uses: WolstenholmePrime.IsWolstenholmePrime, OfNat.ofNat, Nat
Proof uses: id, And, GT.gt, Nat.Prime, Nat.ModEq, HPow.hPow, Nat.choose, HSub.hSub, HMul.hMul, And.intro, Eq, HMod.hMod, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_prime_2, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 6 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem wolstenholme_prime_16483 : IsWolstenholmePrime 16843 := by
…
```
Mentions: WolstenholmePrime.IsWolstenholmePrime
theorem WolstenholmePrime.wolstenholme_prime_2124679 line 62 source
extra axioms: _private.0.WolstenholmePrime.wolstenholme_prime_2124679._native.native_decide.ax_1 what this means
Statement uses: WolstenholmePrime.IsWolstenholmePrime, OfNat.ofNat, Nat
Proof uses: id, And, GT.gt, Nat.Prime, Nat.ModEq, HPow.hPow, Nat.choose, HSub.hSub, HMul.hMul, And.intro, Eq, HMod.hMod, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_prime_2, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4 and 6 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem WolstenholmePrime.wolstenholme_bernoulli Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$ sorry line 71 source
Source documentation
Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$.

depends on sorry what this means
Statement uses: Nat, Iff, WolstenholmePrime.IsWolstenholmePrime, And, GT.gt, OfNat.ofNat, Nat.Prime, Dvd.dvd, Int, Rat.num, bernoulli', HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem wolstenholme_bernoulli (p : ℕ) : IsWolstenholmePrime p ↔
    (p > 7) ∧ Nat.Prime p ∧ ↑p ∣ (bernoulli' (p - 3)).num := by
…
```
Mentions: WolstenholmePrime.IsWolstenholmePrime
theorem WolstenholmePrime.wolstenholme_harmonic Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime if it $p^3$ divides the numerator of the harmonic number $H_{p-1}$ sorry line 80 source
Source documentation
Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime if it $p^3$ divides the numerator of the harmonic number $H_{p-1}$.

depends on sorry what this means
Statement uses: Nat, Iff, WolstenholmePrime.IsWolstenholmePrime, And, GT.gt, OfNat.ofNat, Nat.Prime, Dvd.dvd, Int, HPow.hPow, Rat.num, harmonic, HSub.hSub
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 wolstenholme_harmonic (p : ℕ) : IsWolstenholmePrime p ↔
    (p > 7) ∧ Nat.Prime p ∧ ↑(p ^ 3) ∣ (harmonic (p - 1)).num := by
…
```
Mentions: WolstenholmePrime.IsWolstenholmePrime
theorem WolstenholmePrime.wolstenholme_prime_infinite It is conjectured that there are infinitely many Wolstenholme primes sorry line 90 source
Source documentation
It is conjectured that there are infinitely many Wolstenholme primes.
Reference: Wikipedia

L4DEL explanation
It is conjectured that there are infinitely many Wolstenholme primes. Reference: Wikipedia
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, WolstenholmePrime.IsWolstenholmePrime
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 wolstenholme_prime_infinite :
    {p : ℕ | IsWolstenholmePrime p}.Infinite := by
…
```
Mentions: WolstenholmePrime.IsWolstenholmePrime