Fortune's Conjecture
Fortune's Conjecture: Every Fortunate number is prime.
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.
Cite
@misc{cairn-fortune-conjecture,
title = {Fortune's Conjecture},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/fortune-conjecture}},
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
Fortune's Conjecture: Every Fortunate number is prime.
A Fortunate number is the smallest integer such that p_n\\# + m is prime, where p_n\\# denotes the primorial of the -th prime — equivalently, the product of the first primes.
Fortune's Conjecture asserts that every Fortunate number is prime — equivalently, that no Fortunate number is composite.
The conjecture is named after the social anthropologist Reo Fortune, who proposed it. The first few Fortunate numbers are (OEIS A005235); all known values are prime.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.Wikipedia.FortuneConjecture. answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem fortune_conjecture :
answer(sorry) ↔ (∀ n : ℕ, Nat.Prime (fortunateNumber 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. 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.