diff --git a/FormalConjectures/ErdosProblems/888.lean b/FormalConjectures/ErdosProblems/888.lean index 674ee1e105..3da4fead87 100644 --- a/FormalConjectures/ErdosProblems/888.lean +++ b/FormalConjectures/ErdosProblems/888.lean @@ -1,5 +1,5 @@ /- -Copyright 2025 The Formal Conjectures Authors. +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. @@ -19,37 +19,67 @@ import FormalConjectures.Util.ProblemImports /-! # Erdős Problem 888 -*Reference:* [erdosproblems.com/888](https://www.erdosproblems.com/888) +*References:* +- [erdosproblems.com/888](https://www.erdosproblems.com/888) +- [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number + theory. Number theory (Eger, 1996) (1998), 169-180. -/ open Classical Filter namespace Erdos888 - -/-- Condition on the sets `A` appearing in Erdős 888. Namely, let `A` be a subset -of `{1,...,n}` such that if `a ≤ b ≤ c ≤ d ∈ A` and `abcd` square then `ad=bc`. -/ +/-- +Condition on the sets $A$ appearing in Erdős 888. Namely, let $A$ be a subset +of ${1,...,n}$ such that if $a ≤ b ≤ c ≤ d ∈ A$ and $abcd$ square then $ad=bc$. +-/ def RequiredCondition (A : Finset ℕ) (n : ℕ) : Prop := A ⊆ Finset.Ioc 0 n ∧ ∀ᵉ (a ∈ A) (b ∈ A) (c ∈ A) (d ∈ A), a ≤ b → b ≤ c → c ≤ d → IsSquare (a * b * c * d) → a * d = b * c -/-- Proposition that for a specific `n` an `A` with the above defined condition -and cardinality `k` exists. -/ +/-- +Proposition that for a specific $n$ an $A$ with the above defined condition and cardinality $k$ +exists. +-/ def p (n : ℕ) (k : ℕ) : Prop := ∃ A : Finset ℕ, RequiredCondition A n ∧ A.card = k +/-- +What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if +$a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then $ad=bc$? + +This was proved by GPT-5.5 Pro (prompted by Chojecki). +-/ +@[category research solved, AMS 11] +theorem erdos_888 : + (fun n : ℕ ↦ (Nat.findGreatest (p n) n : ℝ)) =Θ[atTop] + (fun n : ℕ ↦ (n : ℝ) * Real.log (Real.log n) / Real.log n) := by + sorry -/-- What is the size of the largest subset `A` of `{1,...,n}` such that if -`a ≤ b ≤ c ≤ d ∈ A` and `abcd` square then `ad=bc` -/ -@[category research open, AMS 11] -theorem erdos_888 : ∀ n, Nat.findGreatest (p n) n = (answer(sorry) : ℕ → ℕ) n := by +/-- +Erdős claims that Sárközy proved that $\lvert A\rvert =o(n)$ (a proof of this +bound is provided by Tao in the comments). +-/ +@[category research solved, AMS 11] +theorem erdos_888.variants.sarkozy : + (fun n ↦ (Nat.findGreatest (p n) n : ℝ)) =o[atTop] (Nat.cast : ℕ → ℝ) := by sorry -/--`|A|=o(n)`. -/ +/-- +The primes show that $\lvert A\rvert \gg n/\log n$ is possible. +-/ @[category research solved, AMS 11] -theorem erdos_888.variants.sarkozy : (fun n ↦ (Nat.findGreatest (p n) n : ℝ)) =o[atTop] (Nat.cast : ℕ → ℝ) := by +theorem erdos_888.variants.primes : + (fun n : ℕ ↦ (Nat.findGreatest (p n) n : ℝ)) ≫ (fun n : ℕ ↦ (n : ℝ) / Real.log n) := by sorry -/-- The primes show that `|A| ≫ n/log n` is possible. -/ +/-- +Cambie and Weisenberg have noted in the comments that the set of semiprimes +also works, showing $(1+o(1))\frac{\log\log n}{\log n}n \leq \lvert A\rvert$ is achievable. +-/ @[category research solved, AMS 11] -theorem erdos_888.variants.primes : (fun n : ℕ ↦ (Nat.findGreatest (p n) n : ℝ )) ≫ (fun n : ℕ ↦ n / (n : ℝ).log) := by +theorem erdos_888.variants.semiprimes : + (fun n : ℕ ↦ (Nat.findGreatest (p n) n : ℝ)) ≫ + (fun n : ℕ ↦ (n : ℝ) * Real.log (Real.log n) / Real.log n) := by sorry + +end Erdos888 diff --git a/FormalConjectures/Subsets/FC100OpenSet1.lean b/FormalConjectures/Subsets/FC100OpenSet1.lean index ffe2a5a39e..551cd20807 100644 --- a/FormalConjectures/Subsets/FC100OpenSet1.lean +++ b/FormalConjectures/Subsets/FC100OpenSet1.lean @@ -221,6 +221,6 @@ end Subsets.FC100OpenSet1 open Lean Meta ProblemAttributes in #eval verifyCategoryCounts Subsets.FC100OpenSet1.problems [ - ("research open", 95), - ("research solved", 5) + ("research open", 94), + ("research solved", 6) ]