Erdős (Erdos) Problem #820: status in 2026

Erdős Problem #820

Let H(n) be the smallest integer l such that there exists k < l with (k^n-1, l^n-1) = 1. Is it true that H(n) = 3 infinitely often? That is, is (2^n-1, 3^n-1) = 1 for infinitely many n (see Erdos820.H_eq_three_iff)?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #820 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_820 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #820 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_820`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 1 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-820,
  title        = {Erdős Problem #820},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-820}},
  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-820.json)](https://cairn-commons.com/problems/erdos-820)

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 $H (n)$ be the smallest integer $l$ such that there exists $k < l$ with $(k^{n} - 1, l^{n} - 1) = 1$ . Is it true that $H (n) = 3$ infinitely often? That is, is $(2^{n} - 1, 3^{n} - 1) = 1$ for infinitely many $n$ (see Erdos820.H_eq_three_iff)?

The request to estimate $H (n)$ is open-ended and not formalised; the variants below state the specific questions that are asked. See also Erdős Problem 770.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«820».

theorem erdos_820 : {n : ℕ | H n = 3}.Infinite

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/820. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_820.variants.log_log_exponent — Is it true that there exists some constant c > 0 such that, for all ε > 0, H(n) > exp(n^(c-ε)/loglog n) for infinitely many n and H(n) < exp(n^(c+ε)/loglog n)…
erdos_820.variants.two_pow_upper — Let K(n) be the smallest k such that (k^n-1, 2^n-1) = 1. Does a similar upper bound hold for K(n)?

References

erdosproblems.com/820
[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
OEIS A263647: the $n$ with $(2^{n} - 1, 3^{n} - 1) = 1$ .

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.