Number theory
c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than
c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
$5,108 bountynobody has started
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
Even one lemma helps. Whoever finishes this later shares the pool with everyone who got them there.
contrib new erdos-238Command line, then a pull request on GitHub.
Nothing has been published against this problem yet.
A first lemma is worth as much as a last one: whoever closes the problem shares the pool with everyone who got them there.
Formal statement
Proof target
import FormalConjectures.ErdosProblems.«238»
import TaskSupport
namespace Bounty
theorem target : fcTypeOfName% "Erdos238.erdos_238" := by
sorry
end Bounty
Original conjecture (Lean type)
True ↔
∀ c₁ > 0,
∀ c₂ > 0,
∀ᶠ (x : ℝ) in Filter.atTop,
∃ k,
c₁ * Real.log x < ↑k ∧
∃ f m,
(∀ (i : Fin k), ↑(f i) ≤ x ∧ f i = Nat.nth Nat.Prime (m + ↑i)) ∧
∀ (i : Fin (k - 1)), c₂ < primeGap (m + ↑i)Original conjecture source: FormalConjectures/ErdosProblems/238.lean
References
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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