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mo271 merged 15 commits into from
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feat(ErdosProblems): 260 #1629

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/-
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 FormalConjectures.Util.ProblemImports

/-!
# Erdős Problem 260

*Reference:* [erdosproblems.com/260](https://www.erdosproblems.com/260)
-/

namespace Erdos260

open Filter

/-
Let $a_1 < a_2 < \cdots$ be an increasing sequence such that
\frac{a_n}{n} \to \infty.
Is the sum
\sum_{n}^{\infty} \frac{a_n}{2^{a_n}} irrational?
-/

@[category research open, AMS 11]
theorem erdos_260 (a : ℕ → ℝ)(s : ℝ)
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@mo271 mo271 Mar 1, 2026

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Haven't closely looked at the statement, but is it really that the a n are Real not Nat or Int?

In any case, one can proof the negation of this statement found by AlphaProof:

theorem erdos_260_negation: ∃ (a : ℕ → ℝ)(s : ℝ)
                  (h : StrictMono a)
                  (h2 : Tendsto (fun n => a n / (n : ℝ)) atTop atTop)
                  (h3 : HasSum (fun n => a n / 2 ^ (a n)) s),
                  ¬ Irrational s := by
  by_cases h:∃p : ℝ,0<p∧∑'a : ℕ,(a*(a+1): ℝ)/2^(a*(a+1):ℝ)=p
  · convert h.elim fun and x =>(exists_rat_btwn x.1).elim fun A B=>_
    by_cases h:∃p : ℝ,0<p∧∑' (n : ℕ),(n* (n + 1)+p) / 2^(n* (n + 1)+p)=A
    · choose _ _ _simpa using(id) h
      by_cases h:Filter.Tendsto (fun (n : ℕ)=>(n* (n + 1)+by valid: ℝ)/n) .atTop .atTop
      · exact ⟨ _,A, fun and R M=>by gcongr,h,_simpa▸(not_imp_comm.1 tsum_eq_zero_of_not_summable (by bound)).hasSum, not_not_intro ⟨A,by simp_all⟩⟩
      · cases h ((Filter.tendsto_atTop_mono' _) ((Filter.eventually_ne_atTop 0).mono fun and m=>le_of_lt (by field_simp[lt_div_iff₀', *])) (tendsto_natCast_atTop_atTop.comp (Filter.tendsto_add_atTop_nat (1))))
    simp_all[div_eq_mul_inv, add_mul,Real.rpow_add,←inv_pow,tsum_mul_left]
    rw[forall₂_congr fun R M=>not_congr (Eq.congr_left (tsum_add _ _))] at h
    · simp_all only[mul_left_comm (_*_ : ℝ),tsum_mul_left,Real.rpow_def_of_pos two_pos,←Real.exp_neg]
      have := (Real.tendsto_exp_neg_atTop_nhds_zero.comp (Filter.tendsto_id.const_mul_atTop ( Real.log_pos one_lt_two))).mul_const and
      have:=this.add ((tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 (.log (@2)) (by positivity) ).mul_const (∑' (n : ℕ),(( -(Real.log @2* (n* (n + 1)))).exp)))
      cases((this.congr (by norm_num[mul_assoc])).eventually_lt_const (by simp_all)).and (Filter.mem_atTop _)|>.exists.elim fun and x =>(intermediate_value_Ioo' x.2 (by fun_prop) ⟨x.1,by simp_all⟩).elim fun and=>And.elim (h and ·.1)
    · use fun and n=>((not_imp_comm.1 tsum_eq_zero_of_not_summable (x.2▸x.1).ne').mul_left _).congr fun and=>mul_left_comm _ _ _
    · exact (fun R M=>mod_cast(((summable_geometric_two.comp_injective ( strictMono_nat_of_lt_succ (by bound):StrictMono fun and=>and*(and+1)).injective).congr (by simp_all)).mul_left _).mul_left R)
  · apply h.elim ⟨ _,mod_cast(le_tsum ↑_ (1) (by {bound})).trans_lt' (by(norm_num)), rfl⟩
    norm_num[((hasSum_coe_mul_geometric_of_norm_lt_one (by bound:norm (1/2: ℝ)<1)).summable.comp_injective ( strictMono_nat_of_lt_succ (by bound):StrictMono fun and=>and*(and+1)).injective).congr,div_eq_mul_inv,←inv_pow]

so potentially there is a misformalisation here? (Only compiles in v4.22)

(h : StrictMono a)
(h2 : Tendsto (fun n => a n / (n : ℝ)) atTop atTop)
(h3 : HasSum (fun n => a n / 2 ^ (a n)) s) :
Irrational s := by
sorry

end Erdos260
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