Smallest $m > 0$ such that there are no primes between $nm$ and $n(m+1)$ inclusive.

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import FormalConjecturesUtil

/-!
# Smallest $m > 0$ such that there are no primes between $nm$ and $n(m+1)$ inclusive.

Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.

*References:*
- [A110835](https://oeis.org/A110835)
-/

namespace OeisA110835

open Nat Set

/--
The primary defining sequence `a`.
$a(n)$ is the smallest $m > 0$ such that there are no primes between $n \cdot m$
and $n \cdot (m+1)$ inclusive.
-/
noncomputable def a (n : ℕ) : ℕ :=
  let IsPrimeFreeInterval (m : ℕ) : Prop :=
    ∀ p : ℕ, p.Prime → ¬ (n * m ≤ p ∧ p ≤ n * (m + 1))
  let s : Set ℕ := {m : ℕ | m > 0 ∧ IsPrimeFreeInterval m}
  sInf s

/-- Term theorems verifying the first few values of the sequence against the official OEIS b-file -/
@[category test, AMS 11]
theorem a_1 : a 1 = 8 := by
  dsimp [a]
  have hleast : IsLeast {m : ℕ | m > 0 ∧ ∀ p : ℕ, p.Prime → ¬ (1 * m ≤ p ∧ p ≤ 1 * (m + 1))} 8 := by
    constructor
    · simp only [mem_ofPred_eq, one_mul]
      refine ⟨by decide, ?_⟩
      intro p hp ⟨hge, hle⟩
      interval_cases p <;> revert hp <;> decide
    · rintro m ⟨hmpos, hfree⟩
      by_contra! hlt
      simp only [one_mul] at hfree
      interval_cases m
      · specialize hfree 2 (by decide); revert hfree; decide
      · specialize hfree 2 (by decide); revert hfree; decide
      · specialize hfree 3 (by decide); revert hfree; decide
      · specialize hfree 5 (by decide); revert hfree; decide
      · specialize hfree 5 (by decide); revert hfree; decide
      · specialize hfree 7 (by decide); revert hfree; decide
      · specialize hfree 7 (by decide); revert hfree; decide
  exact hleast.csInf_eq

@[category test, AMS 11]
theorem a_2 : a 2 = 4 := by
  dsimp [a]
  have hleast : IsLeast {m : ℕ | m > 0 ∧ ∀ p : ℕ, p.Prime → ¬ (2 * m ≤ p ∧ p ≤ 2 * (m + 1))} 4 := by
    constructor
    · simp only [mem_ofPred_eq]
      refine ⟨by decide, ?_⟩
      intro p hp ⟨hge, hle⟩
      interval_cases p <;> revert hp <;> decide
    · rintro m ⟨hmpos, hfree⟩
      by_contra! hlt
      interval_cases m
      · specialize hfree 2 (by decide); revert hfree; decide
      · specialize hfree 5 (by decide); revert hfree; decide
      · specialize hfree 7 (by decide); revert hfree; decide
  exact hleast.csInf_eq

@[category test, AMS 11]
theorem a_3 : a 3 = 8 := by
  dsimp [a]
  have hleast : IsLeast {m : ℕ | m > 0 ∧ ∀ p : ℕ, p.Prime → ¬ (3 * m ≤ p ∧ p ≤ 3 * (m + 1))} 8 := by
    constructor
    · simp only [mem_ofPred_eq]
      refine ⟨by decide, ?_⟩
      intro p hp ⟨hge, hle⟩
      interval_cases p <;> revert hp <;> decide
    · rintro m ⟨hmpos, hfree⟩
      by_contra! hlt
      interval_cases m
      · specialize hfree 3 (by decide); revert hfree; decide
      · specialize hfree 7 (by decide); revert hfree; decide
      · specialize hfree 11 (by decide); revert hfree; decide
      · specialize hfree 13 (by decide); revert hfree; decide
      · specialize hfree 17 (by decide); revert hfree; decide
      · specialize hfree 19 (by decide); revert hfree; decide
      · specialize hfree 23 (by decide); revert hfree; decide
  exact hleast.csInf_eq

/--
Sierpinski's conjecture (1958) is precisely that $a(n) >= n$ for all $n$.
-/
@[category research open, AMS 11]
theorem conjecture : ∀ n > 0, a n ≥ n := by
  sorry

end OeisA110835

• Formalization source ↗

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