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import FormalConjecturesUtilWolstenholme Prime
namespace WolstenholmePrime
Wolstenholme's theorem states that any prime $p > 3$ satisfies $\binom{2p-1}{p-1} \equiv 1 (\pmod{p^3})$.
Formal proof linked here provided by AlphaProof.
@[category textbook, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/d833ed31d82693f10bed7a4c9ac329545b556a03/FormalConjectures/Wikipedia/WolstenholmePrime.lean#L34"]
theorem wolstenholme_theorem (p : ℕ) (h : p > 3) (hp : Nat.Prime p) :
(2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 3] := p:ℕh:p > 3hp:Nat.Prime p⊢ (2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 3]
All goals completed! 🐙
A prime $p > 7$ is called a
def IsWolstenholmePrime (p : ℕ) : Prop :=
p > 7 ∧ p.Prime ∧ (2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 4]
Two known Wolstenholme primes: 16843 and 2124679.
Formal proof linked here provided by AlphaProof
@[category test, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/d833ed31d82693f10bed7a4c9ac329545b556a03/FormalConjectures/Wikipedia/WolstenholmePrime.lean#L73"]
theorem wolstenholme_prime_16483 : IsWolstenholmePrime 16843 := ⊢ IsWolstenholmePrime 16843
⊢ 16843 > 7 ∧ Nat.Prime 16843 ∧ (2 * 16843 - 1).choose (16843 - 1) % 16843 ^ 4 = 1 % 16843 ^ 4
refine ⟨⊢ 16843 > 7 All goals completed! 🐙, ⊢ Nat.Prime 16843 All goals completed! 🐙, ?_⟩
All goals completed! 🐙
@[category test, AMS 11]
theorem wolstenholme_prime_2124679 : IsWolstenholmePrime 2124679 := ⊢ IsWolstenholmePrime 2124679
⊢ 2124679 > 7 ∧ Nat.Prime 2124679 ∧ (2 * 2124679 - 1).choose (2124679 - 1) % 2124679 ^ 4 = 1 % 2124679 ^ 4
refine ⟨⊢ 2124679 > 7 All goals completed! 🐙, ⊢ Nat.Prime 2124679 All goals completed! 🐙, ?_⟩
All goals completed! 🐙
Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$.
@[category textbook, AMS 11]
theorem wolstenholme_bernoulli (p : ℕ) : IsWolstenholmePrime p ↔
(p > 7) ∧ Nat.Prime p ∧ ↑p ∣ (bernoulli' (p - 3)).num := p:ℕ⊢ IsWolstenholmePrime p ↔ p > 7 ∧ Nat.Prime p ∧ ↑p ∣ (bernoulli' (p - 3)).num
All goals completed! 🐙
Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime if it $p^3$ divides the numerator of the harmonic number $H_{p-1}$.
@[category textbook, AMS 11]
theorem wolstenholme_harmonic (p : ℕ) : IsWolstenholmePrime p ↔
(p > 7) ∧ Nat.Prime p ∧ ↑(p ^ 3) ∣ (harmonic (p - 1)).num := p:ℕ⊢ IsWolstenholmePrime p ↔ p > 7 ∧ Nat.Prime p ∧ ↑(p ^ 3) ∣ (harmonic (p - 1)).num
All goals completed! 🐙
It is conjectured that there are infinitely many Wolstenholme primes.
@[category research open, AMS 11]
theorem wolstenholme_prime_infinite :
{p : ℕ | IsWolstenholmePrime p}.Infinite := ⊢ {p | IsWolstenholmePrime p}.Infinite
All goals completed! 🐙
end WolstenholmePrime