1. FormalConjectures
  2. ErdosProblems
  3. «253»
/- Copyright 2025 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 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ import FormalConjecturesUtil

Erdős Problem 253

Reference: erdosproblems.com/253

namespace Erdos253 open scoped Topology

The predicate that a : ℕ → ℕ is a strictly monotone sequence such that every infinite arithmetic progression contains infinitely many integers that are the sum of distinct $a_i$s.

@[inline] def RepresentsAPs (a : ℕ → ℕ) : Prop := StrictMono a ∧ ∀ l, l.IsAPOfLength ⊤ → (subsetSums (Set.range a) ∩ l).Infinite

Let $a_1 < a_2 < \dotsc$ be an infinite sequence of positive integers such that $\frac{a_{i+1}}{a_i} \to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct $a_i$ then every sufficiently large integer is the sum of distinct $a_i$.

@[category research solved, AMS 11] theorem declaration uses 'sorry'erdos_253 : ¬ ∀ a : ℕ → ℕ, 0 < a 0 → RepresentsAPs a → (Filter.atTop.Tendsto (fun n ↦ (a <| n + 1 : ℝ) / a n) (𝓝 1)) → subsetSums (Set.range a) ∈ Filter.cofinite := ⊢ ¬∀ (a : ℕ → ℕ), 0 < a 0 → RepresentsAPs a → Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) Filter.atTop (𝓝 1) → subsetSums (Set.range a) ∈ Filter.cofinite All goals completed! 🐙 end Erdos253