Firoozbakht's conjecture

formal-conjectures · wikipedia · ams-11

**Firoozbakht's conjecture**
The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$.

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://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.primepuzzles.net/conjectures/conj_030.htm
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Firoozbakht.lean
Local target: Corpus.WikipediaFiroozbakht.firoozbakht_conjecture
Source SHA-256: 8ccc189968c2bc739b276ab194d634732da6cae660d601521c005be620eb9c3b
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.

Public JSON record