2 より大きいすべての偶数は、2つの素数の和で表せるか？（ゴールドバッハ予想）

Determine whether every even integer n > 2 can be written as n = p + q with p and q prime.

The ternary (odd) version is recorded as solved in formal-conjectures (Helfgott).

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

  Every even integer greater than 2 is the sum of two primes, 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 universal quantifier over even integers
  • Exactly two prime summands

  加えられた仮定・条件

  • The bound 'greater than 2' made explicit

  Statement only; no proof exists in the repository.

• The circle method

  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

  Estimate the number of representations of an even N as p + q by major and minor arc analysis of exponential sums over primes.問いと同値

  保たれているもの

  • The exact representation count, if the error term can be controlled

  弱まっているもの

  • For two primes the minor-arc error is not smaller than the main term, so the method yields 'almost all even numbers' rather than 'all'

  加えられた仮定・条件

  • Nothing arithmetic; the cost is analytic control that is currently unavailable

  The same method settles the ternary problem: Vinogradov (1937) for large odd N, Helfgott (2013) for every odd N > 7.

  閉じた道

  • The circle method controls the binary problem only up to an exceptional set: Montgomery-Vaughan (1975) give density zero, not emptiness. Closing that gap is the whole difficulty, not a technical remainder.無条件
    • paperdoi:10.4064/aa-27-1-353-370H. L. Montgomery, R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975) 353-370

  証拠

  • preprintarXiv:1312.7748H. A. Helfgott, The ternary Goldbach conjecture is true (2013); accepted for Annals of Mathematics Studies after referee revisions
  • paperdoi:10.4064/aa-27-1-353-370H. L. Montgomery, R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975) 353-370
  • paperI. M. Vinogradov, Representation of an odd number as a sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937) 291-294The ternary problem for sufficiently large odd numbers, by the circle method

• 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

  Bound below the number of representations N = p + q by sieving the shifted primes, as in Chen's theorem.問いより弱い

  保たれているもの

  • A representation of every large even number as a prime plus something with few prime factors

  弱まっているもの

  • Chen's theorem allows the second summand to be a product of two primes; it is not the conjecture

  加えられた仮定・条件

  • Sieve dimensions and level-of-distribution hypotheses

  閉じた道

  • A sieve alone cannot detect primes (Selberg's parity problem): it cannot separate 'p + prime' from 'p + product of two primes', which is exactly the gap Chen's theorem leaves.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

  証拠

  • paperJ. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973) 157-176Every sufficiently large even number is p + q with q prime or a product of two primes

• 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 every even integer as a sum of two primes 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 representation of every even number as a sum of two primes 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/GoldbachConjecture.lean:33
• otherC. Goldbach, letter to L. Euler, 7 June 1742 (letter XLIII of the Euler-Goldbach correspondence)The original conjecture; Euler's reply of 30 June 1742 states the two-prime form used today
• 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/5ee8ac15-b050-4e68-baab-c7ffd4f02c84

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

版

a7c49591930c

ライセンス

CC-BY-4.0

立場

record_only（Atlas は判定しない）