Erdős Problem #695 — open problem in number theory

Erdős Problem #695

Let q_1 < q_2 < ⋯ be a sequence of primes such that q_i + 1 ≡ 1 pmodq_i. Is it true that lim_k → ∞ q_k^1/k = ∞?

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-erdos-695,
  title        = {Erdős Problem #695},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-695}},
  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

Let $q_{1} < q_{2} < \dots$ be a sequence of primes such that $q_{i + 1} \equiv 1 (mod q_{i})$ . Is it true that $k \to \infty lim q_{k}^{1/ k} = \infty ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«695». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_695 : answer(sorry) ↔
    ∀ {q : ℕ → ℕ},
      StrictMono q →
      (∀ i, (q i).Prime) →
      (∀ i, q (i + 1) % q i = 1) →
      Tendsto (fun k => (q k : ℝ) ^ (1 / k : ℝ)) atTop atTop

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/695. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_695.variants.upperBound — Is there a sequence of primes q_1 < q_2 < ⋯ such that q_i + 1 ≡ 1 pmodq_i and q(k) ≤ exp(k (log k)^1 + o(1))?

References

erdosproblems.com/695

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.