# P3: Goldbach conjecture

- ID: `P3`
- Reference: `goldbach-conjecture`
- Page: https://theoremdb.org/statements/P3
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (A prime number).** A prime number is an integer greater than 1 with exactly two positive divisors.
- **Definition (The claim).** The claim is also called the strong or binary Goldbach conjecture.
- **Remark.** The conjecture has extensive computational support and strong partial results in additive prime number theory.

### What 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.

## Status

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](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- The statement and status were checked against the cited paper on 2026-07-22.
- Helfgott proved the ternary Goldbach conjecture. That theorem concerns sums of three primes and leaves this binary statement unresolved.

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

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- Verification through any finite bound leaves the universal claim open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `goldbach-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2501.02529
   - Also cited at abstract and introduction's Goldbach formulation
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2501.02529, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1090/S0025-5718-2013-02787-1
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Supplies the checked finite-verification frontier without proving the universal statement.
3. <a id="reference-3"></a>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 https://doi.org/10.1142/9789814542487_0019
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides the checked almost-prime neighboring theorem and leaves the required second summand prime.