連続する平方数の間には必ず素数があるか？（ルジャンドル予想）

Determine whether for every integer n ≥ 1 there exists a prime p with n² < p < (n+1)².

One of Landau's four problems on primes. formal-conjectures records a claimed proof for all sufficiently large n (Ferreira, arXiv:2307.08725) as solved; the statement for every n remains open there.

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 n >= 1 there is a prime strictly between n^2 and (n+1)^2, stated in Lean 4 with the answer left open.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 universal quantifier over n
  • The open interval (n^2, (n+1)^2)
  • Primality of the witness

  加えられた仮定・条件

  • An explicit lower bound n >= 1
  • An `answer(sorry)` slot, so the formal statement asks which of 'holds' or 'fails' is provable

  The formal statement is the question, not an attempt at it: no proof exists in the repository.

• 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 every interval (n^2, (n+1)^2), i.e. a gap bound of order p^{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 between consecutive squares 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

• Cramer-style probabilistic model

  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

  Model the primes as independent random events with density 1/log n and read off a prime between consecutive squares as an overwhelmingly likely consequence.問いとずれている

  保たれているもの

  • Predicts the same answer as the conjecture, and matches the computed data

  弱まっているもの

  • Predicts, never proves: a probabilistic model cannot settle an arithmetic statement

  加えられた仮定・条件

  • Independence between prime events, which is false in detail

  閉じた道

  • Maier's theorem shows the Cramer model gives the wrong prediction for primes in intervals of length (log x)^lambda, so its predictions in short intervals cannot be trusted as evidence.無条件
    • paperdoi:10.1307/mmj/1029003189H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221-225: the Cramer model predicts the wrong count in intervals of length (log x)^lambda

• repositoryhttps://github.com/google-deepmind/formal-conjectures@fc2696b2f863e642ef1452938b0f031f5af5bed3FormalConjectures/Wikipedia/LegendreConjecture.lean:34
• talkE. Landau, Geloeste und ungeloeste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion, Proc. 5th International Congress of Mathematicians, Cambridge (1913) 93-108The 1912 ICM address listing the four problems, including this one
• paperdoi:10.5802/jtnb.676J. Pintz, Landau's problems on primes, J. Theor. Nombres Bordeaux 21 (2009) 357-404: modern survey of what is known

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https://atlasalt.com/q/57468fc1-de7b-4daf-a5cd-bb43f29fbf70

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2026-09-14

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2026-09-17

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2027-09-17

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record_only（Atlas は判定しない）