Erdős (Erdos) Problem #1160: status in 2026

Erdős Problem #1160

Let g(n) denote the number of groups of order n. If n≤ 2^m then g(n)≤ g(2^m). This is [Va99, 5.71] and an open problem (Question 22.16) in [BNV07], where its origin could not be traced. It has been attributed to Erdős, Higman and others. Pantelidakis [Pa03] proved it for odd n and m≥ 3619.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-01-23).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1160 as open (status updated 23 January 2026).
Formal Conjectures marks the Lean statement erdos_1160 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1160 on erdosproblems.com	open 2026-01-23	erdosproblems.com
Formal Conjectures: status of `erdos_1160`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 0 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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-erdos-1160,
  title        = {Erdős Problem #1160},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1160}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-1160.json)](https://cairn-commons.com/problems/erdos-1160)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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 $g (n)$ denote the number of groups of order $n$ . If $n \leq 2^{m}$ then $g (n) \leq g (2^{m})$ .

This is [Va99, 5.71] and an open problem (Question 22.16) in [BNV07], where its origin could not be traced. It has been attributed to Erdős, Higman and others. Pantelidakis [Pa03] proved it for odd $n$ and $m \geq 3619$ .

It is a conjecture, not a question. We take $n \geq 1$ ; for $n = 0$ it holds trivially, since $g (0) = 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1160».

theorem erdos_1160 : ∀ n m : ℕ, 0 < n → n ≤ 2 ^ m → g n ≤ g (2 ^ m)

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/1160. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_1160.variants.sum — Question 22.18 of [BNV07] suggests the even stronger conjecture Σ_n<2^mg(n) ≤ g(2^m) for all sufficiently large m (perhaps even as soon as m≥ 7).

References

erdosproblems.com/1160
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
[BNV07] Blackburn, S. R., Neumann, P. M. and Venkataraman, G., Enumeration of finite groups. Cambridge Tracts in Mathematics 173, Cambridge University Press (2007), xii+281.
[Pa03] Pantelidakis, I., On the Number of Non-isomorphic Groups of the Same Order. DPhil thesis, University of Oxford (2003).

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.