Erdős (Erdos) Problem #436: status in 2026

Erdős Problem #436

Is Λ(k, 3) finite for all odd k? Lehmer, Lehmer, Mills and Selfridge [LLMS62] proved Λ(3, 3) = 23532. The question is open for odd k ≥ 5.

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 #436 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_436.parts.ii as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #436 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_436.parts.ii`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 10 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-436,
  title        = {Erdős Problem #436},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-436}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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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

Is $Λ (k, 3)$ finite for all odd $k$ ?

Lehmer, Lehmer, Mills and Selfridge [LLMS62] proved $Λ (3, 3) = 23532$ . The question is open for odd $k \geq 5$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«436».

theorem erdos_436.parts.ii : ∀ k ≥ 3, Odd k → Λ k 3 < ⊤

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/436. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/436
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathematique (1980).
[LeLe62] Lehmer, D. H. and Lehmer, Emma, On runs of residues. Proc. Amer. Math. Soc. (1962), 102-106.
[LLMS62] Lehmer, D. H. and Lehmer, E. and Mills, W. H. and Selfridge, J. L., *Machine proof of a theorem on cubic residues*. Math. Comp. (1962), 407-415.
[BiMi63] Bierstedt, R. G. and Mills, W. H., *On the bound for a pair of consecutive quartic residues of a prime*. Proc. Amer. Math. Soc. (1963), 628-632.
[LLM63] Lehmer, D. H. and Lehmer, Emma and Mills, W. H., Pairs of consecutive power residues. Canadian J. Math. (1963), 172-177.
[BLL64] Brillhart, John and Lehmer, D. H. and Lehmer, Emma, *Bounds for pairs of consecutive seventh and higher power residues*. Math. Comp. (1964), 397-407.
[Gr64g] Graham, R. L., On quadruples of consecutive $k$ th power residues. Proc. Amer. Math. Soc. (1964), 196-197.
[Du65] Dunton, M., Bounds for pairs of cubic residues. Proc. Amer. Math. Soc. (1965), 330-332.
[Hi91] Hildebrand, Adolf, On consecutive $k$ th power residues. II. Michigan Math. J. (1991), 241-253.

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.