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import FormalConjecturesUtilErdős Problem 695
open Filter Finset Real
namespace Erdos695
Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$. Is it true that $$ \lim_{k \to \infty} q_k^{1/k} = \infty? $$
@[category research open, AMS 11]
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 := ⊢ True ↔
∀ {q : ℕ → ℕ},
StrictMono q →
(∀ (i : ℕ), Nat.Prime (q i)) → (∀ (i : ℕ), q (i + 1) % q i = 1) → Tendsto (fun k => ↑(q k) ^ (1 / ↑k)) atTop atTop
All goals completed! 🐙
Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $$ q(k) \leq \exp(k (\log k)^{1 + o(1)})? $$
@[category research open, AMS 11]
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)) := ⊢ True ↔
∃ q,
StrictMono q ∧
(∀ (i : ℕ), Nat.Prime (q i)) ∧
(∀ (i : ℕ), q (i + 1) % q i = 1) ∧
∃ o, o =o[atTop] 1 ∧ ∀ (k : ℕ), ↑(q k) ≤ rexp ((↑k + 1) * log (↑k + 1) ^ (1 + o k))
All goals completed! 🐙
end Erdos695