FormalConjectures.Wikipedia.LegendreConjecture

FormalConjectures
Wikipedia
LegendreConjecture

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import FormalConjecturesUtil

Legendre's conjecture

References:

Landau Problems Wikipedia Page

Legendre Conjecture Wikipedia Page

Luan Alberto Ferreira, Real exponential sums over primes and prime gaps

open Filternamespace LegendreConjecture

Does there always exist at least one prime between consecutive perfect squares?

@[category research open, AMS 11]
theorem declaration uses 'sorry'legendre_conjecture :
    answer(sorry) ↔ ∀ n ≥ 1, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := by⊢ True ↔ ∀ n ≥ 1, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p
  sorryAll goals completed! 🐙

If there exists a constant c > 0 such that (n + 1).nth Nat.Prime - n.nth Nat.Prime < (n.nth Nat.Prime) ^ (1 / 2 - c) for all large n, then Legendre's conjecture is asymptotically true.

Formal proof linked here provided by AlphaProof.

@[category research solved, AMS 11, formal_proof using formal_conjectures at "https://github.com/mzhorvath1/formal-conjectures/blob/a4568d467b4f42884b6a4bd09c40d65f92113ee7/FormalConjectures/Wikipedia/LegendreConjecture.lean#L48"]
theorem declaration uses 'sorry'bounded_gap_legendre
    (H : ∃ c > 0, ∀ᶠ n in atTop, (n + 1).nth Nat.Prime - n.nth Nat.Prime <
      (n.nth Nat.Prime : ℝ) ^ (1 / (2 : ℝ) - c)) :
    ∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := byH:∃ c > 0,
  ∀ᶠ (n : ℕ) in atTop, ↑(Nat.nth Nat.Prime (n + 1)) - ↑(Nat.nth Nat.Prime n) < ↑(Nat.nth Nat.Prime n) ^ (1 / 2 - c)⊢ ∀ᶠ (n : ℕ) in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p
  sorryAll goals completed! 🐙

Ferreira proved that the conjecture is true for sufficiently large n.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'legendre_conjecture.ferreira_large_n :
    ∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p := by⊢ ∀ᶠ (n : ℕ) in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p
  sorryAll goals completed! 🐙


end LegendreConjecture