FormalConjectures.ErdosProblems.«443»

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import FormalConjecturesUtil

Erdős Problem 443

References:

erdosproblems.com/443

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[He25] Hegyvári, Norbert, An elementary question of Erdős and Graham. arXiv:2503.24201 (2025).

namespace Erdos443

The set ${k(m-k) : 1\leq k\leq m/2}$, where m / 2 is ℕ floor division.

def A (m : ℕ) : Finset ℕ := (Finset.Icc 1 (m / 2)).image fun k => k * (m - k)

Let $m,n\geq 1$. What is $$# { k(m-k) : 1\leq k\leq m/2} \cap { l(n-l) : 1\leq l\leq n/2}?$$ Can it be arbitrarily large?

This was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if $m>n$ then the set in question has size $$\leq m^{O(1/\log\log m)},$$ and that for any integer $s$ there exist infinitely many pairs $(m,n)$ such that the set in question has size $s$.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos443.lean"]
theorem erdos_443.parts.i : answer(True) ↔
    ∀ s : ℕ, ∃ m n : ℕ, n < m ∧ s ≤ (A n ∩ A m).card := by⊢ True ↔ ∀ (s : ℕ), ∃ m, ∃ n < m, s ≤ (A n ∩ A m).card
  sorryAll goals completed! 🐙

Let $m,n\geq 1$. What is $$# { k(m-k) : 1\leq k\leq m/2} \cap { l(n-l) : 1\leq l\leq n/2}?$$ Is it $\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos443.lean"]
theorem erdos_443.parts.ii : answer(True) ↔
    ∀ ε : ℝ, 0 < ε → ∃ n₀ : ℕ, ∀ m n : ℕ, n₀ < n → n < m →
      ((A n ∩ A m).card : ℝ) < ((m : ℝ) * n) ^ ε := by⊢ True ↔ ∀ (ε : ℝ), 0 < ε → ∃ n₀, ∀ (m n : ℕ), n₀ < n → n < m → ↑(A n ∩ A m).card < (↑m * ↑n) ^ ε
  sorryAll goals completed! 🐙

end Erdos443