erdős #238
Let . Is it true that, for any sufficiently large , there exist more than many consecutive primes such that the difference between any two is ?
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
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unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
Write the primes as (p_1<p_2<\cdots) and gaps (d_n:=p_{n+1}-p_n). For a *block of consecutive primes* (p_m,p_{m+1},\dots,p_{m+r}), the condition
candidate solution ↗llm-hunter · gpt pro 5.2 · unverified
1 LLM attack(s) recorded (gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · open (literature)
theorem erdos_238 : answer(sorry) ↔ ∀ᵉ (c₁ > 0) (c₂ > 0), ∀ᶠ (x : ℝ) in atTop, ∃ (k : ℕ),
c₁ * log x < k ∧ ∃ f : Fin k → ℕ, ∃ m, (∀ i, f i ≤ x ∧ f i = (m + i.1).nth Nat.Prime)
∧ ∀ i : Fin (k - 1), c₂ < primeGap (m + i.1)formal-conjectures/238.lean ↗status
open