Erdős (Erdos) Problem #1122: status in 2026

Erdős Problem #1122

Let f:ℕ→ ℝ be an additive function (i.e. f(ab)=f(a)+f(b) whenever (a,b)=1). Let A= n ≥ 1: f(n+1)< f(n). If lvert A∩ [1,X]rvert =o(X) then must f(n)=clog n for some c∈ ℝ? The condition lvert A∩ [1,X]rvert =o(X) says that A has natural density 0.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-12-30).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1122 as open (status updated 30 December 2025).
Formal Conjectures marks the Lean statement erdos_1122 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1122 on erdosproblems.com	open 2025-12-30	erdosproblems.com
Formal Conjectures: status of `erdos_1122`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-1122,
  title        = {Erdős Problem #1122},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1122}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $f : N \to R$ be an additive function (i.e. $f (ab) = f (a) + f (b)$ whenever $(a, b) = 1$ ). Let $A = {n \geq 1 : f (n + 1) < f (n)} .$ If $∣ A \cap [1, X]∣ = o (X)$ then must $f (n) = c lo g n$ for some $c \in R$ ?

The condition $∣ A \cap [1, X]∣ = o (X)$ says that $A$ has natural density $0$ .

Partial progress was made by Mangerel [Ma22], who proved that this is true for completely additive $f$ if $∣ A \cap [1, X]∣ ≪ \frac{X}{( lo g X ) ^{2 + c}}$ for some $c > 0$ , and if $f (p)$ does not have very large values (in a certain technical sense).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1122».

theorem erdos_1122 : ∀ f : ℕ → ℝ,
    (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
    {n | 1 ≤ n ∧ f (n + 1) < f n}.HasDensity 0 →
    ∃ c : ℝ, ∀ n > 0, f n = c * Real.log n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1122. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1122
[Er46] Erdős, P., On the distribution function of additive functions. Annals of Math. (1946), 1-20.
[Ma22] Mangerel, Alexander P., *Additive functions in short intervals, gaps and a conjecture of Erdős*. Ramanujan J. (2022), 1023--1090.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.