a(n) = (smallest prime > n^2) - n^2

Conjecture: a(n) ≤ 1 + φ(n) for n > 0. This improves on Oppermann's conjecture, which says a(n) < n. - Thomas Ordowski, Dec 17 2014

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-53000,
  title        = {a(n) = (smallest prime > n^2) - n^2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-53000}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

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On the literature board

The problem

The question

Conjecture: $a (n) \leq 1 + ϕ (n)$ for $n > 0$ . This improves on Oppermann's conjecture, which says $a (n) < n$ .

Thomas Ordowski, Dec 17 2014

The difference between the smallest prime strictly greater than $n^{2}$ and $n^{2}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«53000».

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

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

A053000

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.