Number of primes $p$ such that $n^n \le p \le n^n + n^2$

formal-conjectures · oeis · ams-11

The sequence $a(n)$ counts the number of prime numbers in the interval $[n^n, n^n + n^2]$:
$$a(n) = |\{p \text{ prime} \mid n^n \le p \le n^n + n^2\}|$$

Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \le p \le n^n + n^2$?
In this case, that would be stronger than the Schinzel conjecture: "for $m > 1$ there's at least
one prime $p$ such that $m \le p \le m + \log(m)^2$" since $n^2 < \log(n^n)^2 = n^2 \log(n)^2$.

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://oeis.org/A069922
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/69922.lean
Local target: Corpus.OEIS69922.conjecture
Source SHA-256: b82638ad3d7f1aa45cdca38e029f48b12b310af9772675bfc6fdc6ec0fb748b4
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