443.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

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).

Imports (1)

FormalConjecturesUtil

def Erdos443.A The set $\{k(m-k) : 1\leq k\leq m/2\}$, where m / 2 is ℕ floor division line 38 source
Source documentation
The set $\{k(m-k) : 1\leq k\leq m/2\}$, where m / 2 is ℕ floor division.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.image, HMul.hMul, HSub.hSub, Finset.Icc, OfNat.ofNat, HDiv.hDiv
10 instances, coercions and auxiliary constants left out.
```
def A (m : ℕ) : Finset ℕ := (Finset.Icc 1 (m / 2)).image fun k => k * (m - k)
…
```
theorem Erdos443.erdos_443.parts.i Let $m,n\geq 1$ sorry line 52 source
Source documentation
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This was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if $m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 $s$.

depends on sorry what this means
Statement uses: Iff, True, Nat, Exists, And, LT.lt, LE.le, Finset.card, Inter.inter, Finset, Erdos443.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_443.parts.i : answer(True) ↔
    ∀ s : ℕ, ∃ m n : ℕ, n < m ∧ s ≤ (A n ∩ A m).card := by
…
```
Mentions: Erdos443.A
theorem Erdos443.erdos_443.parts.ii Let $m,n\geq 1$ sorry line 68 source
Source documentation
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 $m,n$?
This was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if $m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 $s$.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, Finset.card, Inter.inter, Finset, Erdos443.A, HPow.hPow, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: Erdos443.A