Legendre conjecture: a prime between consecutive positive squares

Prove that for every positive natural number n there is a prime strictly between n² and (n+1)². The lower bound n ≥ 1 matters: the interval between 0 and 1 contains no prime. Both inequalities are strict. The prime predicate is the elementary divisor definition, so this request requires only pinned Lean core. Research status checked 8 October 2026: no accepted general proof identified in the cited literature. This is a proposed formal statement; independent correspondence review is requested. A successful statement check establishes well-formedness, not the conjecture.

(fun prime : Nat → Prop =>
  ∀ n : Nat, 1 ≤ n → ∃ p : Nat, n ^ 2 < p ∧ p < (n + 1) ^ 2 ∧ prime p)
(fun p => 2 ≤ p ∧ ∀ d : Nat, d ∣ p → d = 1 ∨ d = p)

OA_statement : Prop