Cunningham chains — Jones's conjecture

formal-conjectures · arxiv · ams-11

A Cunningham chain is a sequence of primes satisfying either $p_{i+1}=2p_i+1$
(first kind) or $p_{i+1}=2p_i-1$ (second kind). It is conjectured that there
are infinitely many chains of every positive exact length, of both kinds.

A chain has **exact length k** when its first $k$ terms are prime and the
$(k+1)$-th generated term is composite.

Lenny Jones conjectures that for every positive integer $k$, infinitely many
primes start a chain of exact length $k$, for each of the two kinds.

Jones's conjecture (first kind): for every positive integer $k$, there are
infinitely many primes $p$ that start a first-kind Cunningham chain of
exactly length $k$.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1104.1579
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A181697
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A181715
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Arxiv/1104.1579/CunninghamChain.lean
Local target: Corpus.Arxiv11041579CunninghamChain.infinitely_many_firstKind_chains
Source SHA-256: 8abc7d85928c1daa5a031269f3c5b240b1b2171f6d5587ac454cbd37e59a5feb
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

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