Erdős Problem #445 — open problem in number theory

Erdős Problem #445

Is it true that, for any c>1/2, if p is a sufficiently large prime then, for any n≥ 0, there exist a,b∈(n,n+p^c) such that ab≡ 1pmodp? This is discussed in this MathOverflow question [MathOverflow].

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-445,
  title        = {Erdős Problem #445},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-445}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is it true that, for any $c > 1/2$ , if $p$ is a sufficiently large prime then, for any $n \geq 0$ , there exist $a, b \in (n, n + p^{c})$ such that $ab \equiv 1 (mod p)$ ?

This is discussed in this MathOverflow question [MathOverflow].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«445». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_445 :
    answer(sorry) ↔ ∀ c : ℝ, c > 1 / 2 →
      ∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p 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 — check erdosproblems.com/445. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/445
[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.
MathOverflow

Source and licence

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