/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

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-/
module

public import FormalConjecturesUtil

/-!
# Erdős Problem 342

*References:*
- [erdosproblems.com/342](https://www.erdosproblems.com/342)
- [Ben Green's Open Problem 7](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.7)
- [OEIS A002858](https://oeis.org/A002858)
- [Gu04] Guy, Richard K., *Unsolved problems in number theory* (2004), xviii+437.
-/

@[expose] public section

open Nat Set Filter
open scoped Topology

namespace Erdos342

/-- `UniqueUlamSum a n m` means that $m$ has a unique representation as $a(i) + a(j)$
with $i < j < n$. -/
def UniqueUlamSum (a : ℕ → ℕ) (n m : ℕ) : Prop :=
  ∃! p : ℕ × ℕ, p.1 < p.2 ∧ p.2 < n ∧ m = a p.1 + a p.2

/-- `IsUlamSequence a` means that $a$ is the Ulam sequence (OEIS A002858):
$a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, $a(n)$ is the least integer
greater than $a(n-1)$ that has a unique representation as $a(i) + a(j)$
with $i < j < n$. -/
def IsUlamSequence (a : ℕ → ℕ) : Prop :=
  a 0 = 1 ∧ a 1 = 2 ∧
  ∀ n, 2 ≤ n →
    a (n - 1) < a n ∧
    UniqueUlamSum a n (a n) ∧
    ∀ m, a (n - 1) < m → m < a n → ¬ UniqueUlamSum a n m

/-- $a(0) = 1$ by definition. -/
@[category test, AMS 5 11 40]
theorem erdos_342.test.a0 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 0 = 1 := by
  intro a ⟨ha0, _, _⟩; exact ha0

/-- $a(1) = 2$ by definition. -/
@[category test, AMS 5 11 40]
theorem erdos_342.test.a1 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 1 = 2 := by
  intro a ⟨_, ha1, _⟩; exact ha1

/-- $a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$. -/
@[category test, AMS 5 11 40]
theorem erdos_342.test.a2 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 2 = 3 := by
  intro a ⟨ha0, ha1, ha⟩
  obtain ⟨_, ⟨⟨i, j⟩, ⟨hij, hj, hsum⟩, _⟩, _⟩ := ha 2 (by omega)
  interval_cases j <;> simp_all only [lt_one_iff, not_lt_zero]

/-- $a(3) = 4$: among sums $> 3$ with a unique representation from $\{1,2,3\}$,
the smallest is $4 = 1 + 3$. The candidate $5 = 2 + 3$ is ruled out by minimality since
$4$ has a unique representation. -/
@[category test, AMS 5 11 40]
theorem erdos_342.test.a3 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 3 = 4 := by
  intro a ⟨ha0, ha1, ha⟩
  have ha2 := erdos_342.test.a2 a ⟨ha0, ha1, ha⟩
  obtain ⟨hinc, ⟨⟨i, j⟩, ⟨hij, hj, hsum⟩, _⟩, hmin⟩ := ha 3 (by omega)
  simp only [show (3 : ℕ) - 1 = 2 from rfl] at hinc hmin
  -- hinc : a 2 < a 3, hmin : ∀ m, a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 m
  -- hsum : a 3 = a i + a j, hij : i < j, hj : j < 3
  -- Enumerate j ∈ {0, 1, 2}
  interval_cases j
  · -- j = 0: i < 0 impossible
    omega
  · -- j = 1: i = 0, so a 3 = a 0 + a 1 = 1 + 2 = 3, but a 3 > a 2 = 3
    have hi : i = 0 := by omega
    subst hi; rw [ha0, ha1] at hsum; rw [ha2] at hinc; omega
  · -- j = 2
    interval_cases i
    · -- i = 0: a 3 = a 0 + a 2 = 1 + 3 = 4
      rw [ha0, ha2] at hsum; exact hsum
    · -- i = 1: a 3 = a 1 + a 2 = 2 + 3 = 5
      rw [ha1, ha2] at hsum
      -- hsum : a 3 = 5. Use minimality: m = 4 has unique sum, contradiction.
      exfalso
      have h4 := hmin 4 (by rw [ha2]; omega) (by omega)
      apply h4
      -- Goal: UniqueUlamSum a 3 4, i.e. ∃! (p : ℕ × ℕ), p.1 < p.2 ∧ p.2 < 3 ∧ 4 = a p.1 + a p.2
      -- Witness: (0, 2) since a 0 + a 2 = 1 + 3 = 4
      refine ⟨⟨0, 2⟩, ⟨by omega, by omega, by rw [ha0, ha2]⟩, ?_⟩
      -- Uniqueness: check all pairs (i', j') with i' < j' < 3
      rintro ⟨i', j'⟩ ⟨hij', hj', hsum'⟩
      simp only [Prod.mk.injEq]
      interval_cases j'
      · omega
      · interval_cases i'
        · rw [ha0, ha1] at hsum'; omega
      · interval_cases i'
        · rw [ha0, ha2] at hsum'; constructor <;> omega
        · rw [ha1, ha2] at hsum'; omega

/--
Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence? -/
@[category research open, question, AMS 5 11 40]
theorem erdos_342.parts.i :
    ∀ a : ℕ → ℕ, IsUlamSequence a →
      Set.Infinite {n : ℕ | ∃ m, a m = a n + 2} := by
  sorry

/--
Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?
-/
@[category research open, question, AMS 5 11 40]
theorem erdos_342.parts.ii :
    ∀ a : ℕ → ℕ, IsUlamSequence a →
      let d (n : ℕ) : ℤ := a (n + 1) - a n
      ∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m := by
  sorry

/--
Part (iii), is the density of the sequence 0?
-/
@[category research open, question, AMS 5 11 40]
theorem erdos_342.parts.iii :
    ∀ a : ℕ → ℕ, IsUlamSequence a →
      Set.upperDensity (Set.range a) = 0 := by
  sorry

end Erdos342