n 番目の素数の n 乗根は、単調に減少するか？（フィローズバフト予想）

n 番目の素数の n 乗根からなる数列が、n について狭義単調減少かを決定せよ。

Decide whether the sequence given by the n-th root of the n-th prime is strictly decreasing in n.

The conjecture implies prime gap bounds stronger than Cramer's conjecture, which is why several researchers expect it to be false even though it holds for every prime checked so far.

解決したかどうかは、問いではなくアプローチごとに決まります。Atlas は判定しません。外部の判定者が何をしたかを記録します。

• Lean 4 formalization (formal-conjectures)

  形式証明

  到達状況

  未着手

  判定者

  the Lean 4 typechecker, against the pinned toolchain · 型検査

  定式化

  The n-th root of the n-th prime is strictly decreasing, stated in Lean 4.問いと同値Lean v4.33.1 + mathlib@0df444a3 (formal-conjectures@fc2696b2)検証対象を見る

  保たれているもの

  • The monotonicity claim

  加えられた仮定・条件

  • Zero-based indexing of the primes through `Nat.nth Nat.Prime`

  Statement only; the repository also records a consequence for prime gaps as solved.

• Exhaustive computation

  計算による判定

  到達状況

  進行中

  判定者

  execution against maximal prime gap tables · 実行

  定式化

  Check the inequality for every prime up to a bound, using maximal gap data.問いより弱い

  保たれているもの

  • Definitive for the range covered

  弱まっているもの

  • Finitely many primes

  加えられた仮定・条件

  • Trust in the gap tables

  閉じた道

  • A finite check cannot settle the statement; and because the conjecture implies gap bounds stronger than Cramer's, agreement over a finite range is weak evidence.無条件
    • preprintarXiv:1503.01744A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion (2015)

  証拠

  • preprintarXiv:1503.01744A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion (2015)

• Probabilistic models of prime gaps

  専門家の合意

  到達状況

  判定者なし・判定がつかない

  判定者

  none: a heuristic model has no decision procedure · 判定手続きなし

  定式化

  Compare the conjecture with what random models of the primes predict for large gaps.問いとずれている

  保たれているもの

  • Places the conjecture among the known gap conjectures

  弱まっているもの

  • Models predict occasional gaps larger than the conjecture allows, so the heuristic evidence points against it

  加えられた仮定・条件

  • Independence assumptions known to be false in detail

  閉じた道

  • Maier's theorem shows the random model mispredicts primes in short intervals, so neither the model's support nor its objection can be treated as settled evidence.無条件
    • paperdoi:10.1307/mmj/1029003189H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221-225: the Cramer model's predictions fail in short intervals, which is the ground for doubting Firoozbakht-type bounds

• repositoryhttps://github.com/google-deepmind/formal-conjectures@fc2696b2f863e642ef1452938b0f031f5af5bed3FormalConjectures/Wikipedia/Firoozbakht.lean:42
• otherF. Firoozbakht, reported in P. Ribenboim, The Little Book of Bigger Primes (2nd ed., Springer 2004), p. 185The conjecture as first reported

URI

https://atlasalt.com/q/b04badac-ee08-42cf-a74f-ffbe61d7f20b

登録

2026-09-17

最終レビュー

2026-09-19

次回レビュー期限

2027-09-19

版

631e6ccc05f7

ライセンス

CC-BY-4.0

立場

record_only（Atlas は判定しない）