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Level A · Machine-checkable Hard Number theory P-erdos-1072

Erdős Problem #1072

Is it true that there are infinitely many p for which f(p) = p − 1?

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.

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Cite 
@misc{cairn-erdos-1072,
  title        = {Erdős Problem #1072},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1072}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

erdos_1072.parts.i. Is it true that there are infinitely many for which ?

erdos_1072.parts.ii. Is it true that for in a density 1 subset of the primes?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1072» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1072.parts.i : answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ f p = p - 1}
theorem erdos_1072.parts.ii :
    answer(sorry) ↔ ∃ (P : Set ℕ), P ⊆ {p | p.Prime} ∧ P.HasDensity 1 {p | p.Prime} ∧
      Tendsto (fun p => (f p / p : ℝ)) (atTop ⊓ principal P) (𝓝 0)

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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/1072. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_1072.variants.littleo — Erdős, Hardy, and Subbarao [HaSu02], believed that the number of p ≤ x for which f(p)=p−1 is o(x/log x). [HaSu02] Hardy, G. E. and Subbarao, M.

References

erdosproblems.com/1072

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.