p と p+2 がともに素数となる組は無限にあるか？（双子素数予想）

Determine whether the set of primes p for which p + 2 is also prime is infinite.

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

  There are infinitely many primes p with p + 2 also prime, 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

  • Infinitude
  • The exact gap of 2

  加えられた仮定・条件

  • Nothing beyond the Lean encoding of 'infinitely many'

  Statement only; no proof exists in the repository.

• Sieve methods and bounded gaps

  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

  Drive the admissible gap in 'infinitely many prime pairs differing by H' down to H = 2, via the GPY method and its successors.問いより弱い

  保たれているもの

  • Infinitude of prime pairs at a bounded distance

  弱まっているもの

  • Gives some gap H, not the gap 2 the question asks about: H = 246 unconditionally today

  加えられた仮定・条件

  • Admissible tuples and distribution hypotheses of Elliott-Halberstam type

  GPY (2009) -> Zhang (2013, H < 70,000,000) -> Maynard and Polymath8b (2014, H = 246).

  閉じた道

  • A sieve alone cannot detect primes (Selberg's parity problem), so this family reaches almost-primes and bounded gaps, never the gap 2 unaided.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
  • Even assuming the generalized Elliott-Halberstam conjecture, the method reaches H = 6, not H = 2; the route stops short by construction.条件つき（the Elliott-Halberstam conjecture and its generalized form）— 前提が崩れれば道は開く
    • paperarXiv:1407.4897D.H.J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes: gap 246 unconditionally, 12 under Elliott-Halberstam, 6 under its generalized form

  証拠

  • paperarXiv:math/0508185D. Goldston, J. Pintz, C. Yildirim, Primes in tuples I, Ann. of Math. 170 (2009) 819-862
  • paperdoi:10.4007/annals.2014.179.3.7Y. Zhang, Bounded gaps between primes, Ann. of Math. 179 (2014) 1121-1174: some gap below 70 million occurs infinitely often
  • paperarXiv:1311.4600J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015) 383-413
  • paperarXiv:1407.4897D.H.J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes: gap 246 unconditionally, 12 under Elliott-Halberstam, 6 under its generalized form

• Hardy-Littlewood k-tuples heuristic

  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 conjectural asymptotic has no decision procedure · 判定手続きなし

  定式化

  Read the twin prime count off the first Hardy-Littlewood conjecture, pi_2(x) ~ 2 C_2 x / (log x)^2.問いより強い

  保たれているもの

  • Predicts infinitude, and matches the computed counts closely

  弱まっているもの

  • Predicts, never proves

  加えられた仮定・条件

  • The singular-series model of prime correlations

  閉じた道

  • The prime k-tuples conjecture is incompatible with the second Hardy-Littlewood conjecture (Hensley-Richards 1974), so this heuristic family is not internally consistent and one member must be false.無条件
    • paperdoi:10.4064/aa-25-4-375-391D. Hensley, I. Richards, Primes in intervals, Acta Arith. 25 (1974) 375-391: the prime k-tuples conjecture and pi(x+y) <= pi(x) + pi(y) cannot both hold

• 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 twin prime pairs 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:10.1007/BF02403921G. H. Hardy, J. E. Littlewood, Some problems of 'Partitio numerorum' III, Acta Math. 44 (1923) 1-70: the first conjecture, pi_2(x) ~ 2 C_2 x / (log x)^2

• repositoryhttps://github.com/google-deepmind/formal-conjectures@fc2696b2f863e642ef1452938b0f031f5af5bed3FormalConjectures/Wikipedia/TwinPrimes.lean:34
• paperA. de Polignac, Six propositions arithmologiques deduites du crible d'Eratosthene, Nouv. Ann. Math. 8 (1849) 423-429The gap-2 case of Polignac's conjecture
• 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

URI

https://atlasalt.com/q/2ea67b54-58cd-4f29-b234-547543ffcfe6

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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record_only（Atlas は判定しない）