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Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/
import FormalConjecturesUtil
open Real
namespace Firoozbakht
The sequence of $\sqrt[n]{p_n}$ where $p_n$ is the n:th prime number.
noncomputable def firoozbakhtSeq (n : ℕ) : ℝ :=
(n.nth Prime)^(1/(n + 1) : ℝ)
Firoozbakht's conjecture The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$.
@[category research open, AMS 11]
theorem firoozbakht_conjecture (n : ℕ) :
firoozbakhtSeq (n+1) < firoozbakhtSeq n := n:ℕ⊢ firoozbakhtSeq (n + 1) < firoozbakhtSeq n
All goals completed! 🐙
The inequality $p_{n+1}-p_n < (\log p_n)^2-\log p_n$ holds for all $n>4$. A consequence of Firuzbakht's conjecture.
@[category research solved, AMS 11]
theorem firoozbakht_conjecture_consequence (n : ℕ) (hn : 3 < n) (P : type_of% firoozbakht_conjecture) :
(n+1).nth Prime - n.nth Prime < (log (n.nth Prime))^2 - log (n.nth Prime) := n:ℕhn:3 < nP:∀ (n : ℕ), firoozbakhtSeq (n + 1) < firoozbakhtSeq n⊢ ↑(Nat.nth Prime (n + 1)) - ↑(Nat.nth Prime n) < log ↑(Nat.nth Prime n) ^ 2 - log ↑(Nat.nth Prime n)
All goals completed! 🐙
end Firoozbakht