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n² + 1 の形の素数は無限に存在するか？（ランダウの第4問題）

Are there infinitely many primes of the form n squared plus one?

mathematicsnumber-theoryprime-patterns

問題文

n² + 1 が素数となる整数 n が無限に存在するかを決定せよ。

Decide whether infinitely many integers n make n^2 + 1 prime.

背景

The fourth of Landau's problems, and the one where the parity obstruction bites hardest: sieve methods give almost-primes, and the only known way past parity so far required a polynomial in two variables rather than one.

アプローチ · 3 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
    Infinitely many n with n^2+1 prime, 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

    • Infinitude
    • The polynomial n^2+1

    加えられた仮定・条件

    • An `answer(sorry)` slot, so the formal statement asks which of 'holds' or 'fails' is provable

    Statement only.

  • Sieve methods

    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
    Sieve the values of n^2+1 to produce infinitely many primes.問いと同値

    保たれているもの

    • Directly targets the question

    弱まっているもの

    • Iwaniec's theorem gives infinitely many n with n^2+1 a product of at most two primes, not a prime

    加えられた仮定・条件

    • Sieve dimensions and level-of-distribution hypotheses

    閉じた道

    • A sieve alone cannot distinguish a prime from a product of two primes, which is exactly the gap between Iwaniec's theorem and the question.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

    証拠

    • paperdoi:10.4064/aa-24-5-435-459H. Iwaniec, Almost-primes represented by quadratic polynomials, Acta Arith. 24 (1974) 435-459: n^2+1 is a product of at most two primes infinitely often

  • Parity-sensitive methods for polynomials in two variables

    査読付きの証明

    到達状況
    未着手
    判定者
    the analytic number theory community, through journal peer review · 査読
    定式化
    Adapt the bilinear methods that capture primes of the form a^2 + b^4 to the one-variable polynomial n^2 + 1.問いとずれている

    保たれているもの

    • The only known technique that has broken parity for a sparse polynomial family

    弱まっているもの

    • Requires two free variables; n^2+1 has one, and the sparseness is of a different order

    加えられた仮定・条件

    • Bilinear structure that n^2+1 does not have

    閉じた道

    • The known parity-breaking results rely on a second variable, so they do not apply to a single-variable polynomial; no adaptation is known.無条件
      • paperdoi:10.2307/121034J. Friedlander, H. Iwaniec, The polynomial X^2+Y^4 captures its primes, Ann. of Math. 148 (1998) 945-1040

    証拠

    • paperdoi:10.2307/121034J. Friedlander, H. Iwaniec, The polynomial X^2+Y^4 captures its primes, Ann. of Math. 148 (1998) 945-1040

出典

記録

URI
https://atlasalt.com/q/44b821aa-4150-40b2-862f-088ff203f093
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2026-09-17
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2026-09-19
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