FormalConjectures.ErdosProblems.«1061»

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import FormalConjecturesUtil

Erdős Problem 1061

References:

erdosproblems.com/1061

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), Problem B15.


open Filter Asymptotics

open scoped ArithmeticFunction.sigma


namespace Erdos1061

Let S x count the number of ordered pairs of positive integers (a, b) with a + b ≤ x such that σ(a) + σ(b) = σ(a + b), where σ is the sum of divisors function.

In particular, (a, b) and (b, a) are counted separately; an unordered variant could be obtained by additionally requiring a ≤ b.

noncomputable abbrev S (x : ℝ) : ℝ :=
  ((Finset.Icc 1 ⌊x⌋₊ ×ˢ Finset.Icc 1 ⌊x⌋₊).filter fun (a, b) ↦
      a + b ≤ x ∧ σ 1 a + σ 1 b = σ 1 (a + b)).card

How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1061 : answer(sorry) ↔ ∃ c : ℝ, 0 < c ∧ S ~[atTop] (fun x : ℝ ↦ c * x) := by⊢ True ↔ ∃ c, 0 < c ∧ S ~[atTop] fun x => c * x
  sorryAll goals completed! 🐙


end Erdos1061