Andrica's conjecture
Andrica's conjecture
Inferred from the community field; finer classification has not been reviewed.
Problem statement
andrica_conjecture
Andrica's conjecture The inequality holds for all , where is the -th prime number.
Lean 4
Formal statement
Lean 4/-
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-/
import FormalConjecturesUtil
/-!
# Andrica's conjecture
*References:*
- [Wikipedia](https://en.wikipedia.org/wiki/Andrica%27s_conjecture)
- [Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*](https://arxiv.org/abs/2307.08725)
-/
open Filter
namespace Andrica
/--
**Andrica's conjecture**
The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.
-/
@[category research open, AMS 11]
theorem andrica_conjecture (n : ℕ) :
Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1 := by
sorry
/--
Ferreira proved that Andrica's conjecture is true for sufficiently large n.
-/
@[category research solved, AMS 11]
theorem andrica_conjecture.ferreira_large_n :
∀ᶠ n in atTop, Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1 := by
sorry
end Andrica
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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