Euclid Numbers conjecture
It is not known whether there is an infinite number of prime Euclid numbers.
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.
Cite
@misc{cairn-euclid,
title = {Euclid Numbers conjecture},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/euclid}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
infinite_prime_euclid_numbers. It is not known whether there is an infinite number of prime Euclid numbers.
euclid_numbers_are_square_free. It is not known whether every Euclid number is a square-free number.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.Wikipedia.Euclid (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem infinite_prime_euclid_numbers : answer(sorry) ↔ {n | (Euclid n).Prime}.Infinitetheorem euclid_numbers_are_square_free : answer(sorry) ↔ (∀ n, Squarefree (Euclid n))What counts as progress
- A Lean proof of one of the statements above, pinned as the claim's formal statement.
- 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. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
Source and licence
Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.