FormalConjectures.Wikipedia.PierpontPrime

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import FormalConjecturesUtil

Pierpont primes

A Pierpont prime is a prime of the form $2^a 3^b + 1$, where $a$ and $b$ are nonnegative integers. Marc Gleason conjectured that there are infinitely many.

References:

Wikipedia, Pierpont prime

OEIS A005109

namespace PierpontPrime

A Pierpont prime is a prime of the form $2^a3^b + 1$.

def IsPierpontPrime (p : ℕ) : Prop :=
  p.Prime ∧ ∃ a b : ℕ, p = 2 ^ a * 3 ^ b + 1

@[category test, AMS 11]
theorem two_isPierpontPrime : IsPierpontPrime 2 := by⊢ IsPierpontPrime 2
  refine ⟨by⊢ Nat.Prime 2 norm_numAll goals completed! 🐙, 0, 0, ?_⟩
  norm_numAll goals completed! 🐙

@[category test, AMS 11]
theorem thirtySeven_isPierpontPrime : IsPierpontPrime 37 := by⊢ IsPierpontPrime 37
  refine ⟨by⊢ Nat.Prime 37 norm_numAll goals completed! 🐙, 2, 2, ?_⟩
  norm_numAll goals completed! 🐙

There are infinitely many Pierpont primes.

@[category research open, AMS 11]
theorem infinitely_many_pierpont_primes :
    Set.Infinite {p : ℕ | IsPierpontPrime p} := by⊢ {p | IsPierpontPrime p}.Infinite
  sorryAll goals completed! 🐙

end PierpontPrime