Number of primes p such that n^n ≤ p ≤ n^n + n^2

Question: for any n > 0, is there at least one prime p such that n^n ≤ p ≤ 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 ≤ p ≤ m + log(m)^2" since n^2 < log(n^n)^2 = n^2 log(n)^2.

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.

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Cite

@misc{cairn-oeis-69922,
  title        = {Number of primes p such that n^n ≤ p ≤ n^n + n^2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-69922}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«69922».

theorem conjecture (n : ℕ) (hn : 0 < n) : 1 ≤ a 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

A069922

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.