Source labels openChecked July 26, 2026

Erdős Problem 695: Upper Bound

Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $q(k) \leq \exp(k (\log k)^{1 + o(1)})?$

Number theory AMS 11

Mathematical statement

Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and

q(k) \leq \exp(k (\log k)^{1 + o(1)})?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_695.variants.upperBound

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_695.variants.upperBound : answer(sorry) ↔    ∃ q : ℕ → ℕ,      StrictMono q ∧      (∀ i, (q i).Prime) ∧      (∀ i, q (i + 1) % q i = 1) ∧      ∃ o : ℕ → ℝ,        (o =o[atTop] (1 : ℕ → ℝ)) ∧        -- We use `(k + 1)` here as the informal statement is 1-indexed.        ∀ k, q k ≤ exp ((k + 1) * log (k + 1) ^ (1 + o k)) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/695background