[#P3] Goldbach conjecture

Problem. Every even integer \(N>2\) is expressible as \(N=p+q\), where \(p\) and \(q\) are prime.

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Definitions and notation

1Context

The conjecture has extensive computational support and strong partial results in additive prime number theory.

2Problem setup

Definition 1 (A prime number). A prime number is an integer greater than 1 with exactly two positive divisors.

Remark 1. The claim is also called the strong or binary Goldbach conjecture.

3What counts as a solution

Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer greater than 2 and prove that no such prime pair exists.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Oliveira e Silva, Herzog, and Pardi verified the binary Goldbach conjecture for every even integer up to 4 × 10^18. Chen proved that every sufficiently large even integer is the sum of a prime and an integer with at most two prime factors. The binary conjecture for all even integers remains open. Exact unresolved remainder: Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer and prove that no such pair exists.^[1][2][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Oliveira e Silva, Herzog, and Pardi verified the binary Goldbach conjecture for every even integer up to 4 × 10^18. Chen proved that every sufficiently large even integer is the sum of a prime and an integer with at most two prime factors. The binary conjecture for all even integers remains open. Exact unresolved remainder: Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer and prove that no such pair exists.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Oliveira e Silva, Herzog, and Pardi verified the binary Goldbach conjecture for every even integer up to 4 × 10^18. Chen proved that every sufficiently large even integer is the sum of a prime and an integer with at most two prime factors. The binary conjecture for all even integers remains open. Exact unresolved remainder: Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer and prove that no such pair exists.^[1][2][3] Relevance to this problem For Goldbach conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Oliveira e Silva, Herzog, and Pardi verified the binary Goldbach conjecture for every even integer up to 4 × 10^18. Chen proved that every sufficiently large even integer is the sum of a prime and an integer with at most two prime factors. The binary conjecture for all even integers remains open. Exact unresolved remainder: Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer and prove that no such pair exists. Open full record →
Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer greater than 2 and prove that no such prime pair exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer greater than 2 and prove that no such prime pair exists.^[1] Relevance to this problem For Goldbach conjecture, record goldbach-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer greater than 2 and prove that no such prime pair exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every even integer greater than 2 is a sum of two primes, or exhibit an even integer greater than 2 and prove that no such prime pair exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2025 research paper identifies the Goldbach conjecture as open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Helfgott proved the ternary Goldbach conjecture. That theorem concerns sums of three primes and leaves this binary statement unresolved.

Recorded example 1. 4 = 2 + 2, 10 = 3 + 7, and 28 = 5 + 23.

Computational notes

Verification through any finite bound leaves the universal claim open.

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Goldbach conjecture.” TheoremDB. P3. Problem statement; statement text SHA-256 d6d43376a30fb38bf34e12da253bbb787927e609b5882b91b18a4940628d773d. https://theoremdb.org/statement/?ref=P3

BibTeX

@misc{theoremdb-problem-d6d43376a30fb38bf34e12da253bbb787927e609b5882b91b18a4940628d773d,
  title = {{Goldbach conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 d6d43376a30fb38bf34e12da253bbb787927e609b5882b91b18a4940628d773d},
  url = {https://theoremdb.org/statement/?ref=P3}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Shamik Ghosh, “Prime Multiple Missing Graphs”. Prime multiple missing graphs, Arai, K.(eds) Intelligent Computing. CompCom 2025. Lecture Notes in Networks and Systems, vol 1423. Springer, Cham. DOI 10.1007/978-3-031-92602-0_11. arXiv:2501.02529 (2025). Shamik Ghosh, arXiv:2501.02529, abstract and introduction. ↗preprint · primary source · arXiv:2501.02529, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2025 research paper identifies the Goldbach conjecture as open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and introduction's Goldbach formulation.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides a current research source that treats the binary Goldbach statement as open.Source named by the research packet.
Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸”. Mathematics of Computation 83(288) (2013), 2033-2060. DOI 10.1090/S0025-5718-2013-02787-1. complete empirical verification of binary Goldbach through 4 × 10^18. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supplies the checked finite-verification frontier without proving the universal statement.
Jing-Run Chen, “On the representation of a larger even integer as the sum of a prime and the product of at most two primes,” Scientia Sinica 16(2) (1973), 157-176; reprinted in Goldbach Conjecture (World Scientific, 1984), 253-272. DOI 10.1142/9789814542487_0019. main theorem for sufficiently large even integers as a prime plus a P2 integer. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Provides the checked almost-prime neighboring theorem and leaves the required second summand prime.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P3] Goldbach conjecture

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2Problem setup

3What counts as a solution

1Status

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Computational notes

2See also

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Computational notes

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