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OP-9293OpenNumber theory

Erdős #985 · number theory

Is it true that, for every prime pp, there is a prime q<pq<p which is a primitive root modulo pp?

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓

Problem statement

Is it true that, for every prime pp, there is a prime q<pq<p which is a primitive root modulo pp?

Lean 4

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Formal statement

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Proofs & verification

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Background

Artin conjectured that 22 is a primitive root for infinitely many primes pp, which Hooley [Ho67b] proved assuming the Generalised Riemann Hypothesis. Heath-Brown [He86b] proved that at least one of 22, 33, or 55 is a primitive root for infinitely many primes pp.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #985 link
  2. [2]Upstream Lean 4 statement for Erdős #985 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 985 link

Recorded progress

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Discussion

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