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x ≥ 2 のとき、x(x−1) と x² の間、x² と x(x+1) の間にそれぞれ素数があるか？（オッペルマン予想）

For every x ≥ 2, is there a prime in (x(x−1), x²) and another in (x², x(x+1))?

mathematicsnumber-theoryprime-gaps

問題文

Determine whether for every integer x ≥ 2 there is a prime strictly between x(x−1) and x², and a prime strictly between x² and x(x+1).

背景

Stronger than both Legendre's and Brocard's conjectures; both implications are proved in Lean in formal-conjectures.

アプローチ · 4 件

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
    For every x >= 2 there is a prime in (x(x-1), x^2) and another in (x^2, x(x+1)), 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

    • Both half-intervals around x^2
    • The universal quantifier over x >= 2

    加えられた仮定・条件

    • An explicit lower bound x >= 2

    Statement only; the repository carries proofs of the implications to Legendre and Brocard, not of the conjecture.

  • Analytic methods for primes in short intervals

    査読付きの証明

    到達状況
    進行中
    判定者
    the analytic number theory community, through journal peer review · 査読
    定式化
    Prove an unconditional upper bound on prime gaps strong enough to force a prime in each half-interval around x^2, i.e. a gap bound of order x^{1/2}.問いより強い

    保たれているもの

    • The statement is derived, not assumed: a gap bound of the right order settles the question for all large arguments

    加えられた仮定・条件

    • Leaves small cases to computation
    • Proves far more than the question asks, which is why the route is hard

    Best unconditional result: primes in [x - x^0.525, x] for large x (Baker-Harman-Pintz 2001), short of the x^0.5 these questions need.

    閉じた道

    • Assuming the Riemann Hypothesis yields only p_{n+1} - p_n = O(sqrt(p_n) log p_n), which is larger than the interval these questions provide; the route 'assume RH and conclude' is closed.無条件
      • paperH. Cramer, Some theorems concerning prime numbers, Arkiv for Matematik, Astronomi och Fysik 15 (1920), no. 5, 1-32Assuming the Riemann Hypothesis, p_{n+1} - p_n = O(sqrt(p_n) log p_n)

  • Sieve methods

    閉じた道（障壁）

    到達状況
    この道では到達できないと証明済み
    判定者
    none: the obstruction is a published theorem, not a decision procedure · 引用

    差分（このアプローチが問いの何を保ち、何を弱めるか）はまだ書かれていません。人類審査で書きます。

    閉じた道

    • A sieve alone cannot detect primes (Selberg's parity problem), so sieve methods reach almost-primes rather than a prime in a short interval.無条件
      • talkA. Selberg, On elementary methods in prime number theory and their limitations, Den 11te Skandinaviske Matematikerkongress, Trondheim (1949), 13-22The parity obstruction: a sieve cannot distinguish an odd from an even number of prime factors

  • Exhaustive computation

    計算による判定

    到達状況
    進行中
    判定者
    execution against published tables, reproducible by rerunning the search · 実行
    定式化
    Verify a prime in each of (x(x-1), x^2) and (x^2, x(x+1)) for every case up to a finite bound.問いより弱い

    保たれているもの

    • The exact arithmetic claim, case by case

    弱まっているもの

    • Only finitely many cases; the question is about all of them

    加えられた仮定・条件

    • Trust in the search code and in the prime tables it consumes

    閉じた道

    • A finite search cannot settle a statement quantified over all integers; it can only refute it by finding a counterexample.無条件
      • paperdoi:10.1090/S0025-5718-2013-02787-1T. Oliveira e Silva, S. Herzog, S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4x10^18, Math. Comp. 83 (2014) 2033-2060

つながり

この問いが肯定的に解決すれば、成り立つ問い

出典

記録

URI
https://atlasalt.com/q/80e31128-38f8-41ee-9a63-aa2b54e2a785
登録
2026-09-14
最終レビュー
2026-09-17
次回レビュー期限
2027-09-17
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d2cae96fc25c
ライセンス
CC-BY-4.0
立場
record_only（Atlas は判定しない）