Additive prime number theory

Goldbach's Conjecture

Collaboration beta

Every even integer at least four is conjectured to be the sum of two primes. This source turns that question into positivity of an exact weighted count and narrows its preferred analytic route to a difficult multiplier-collar estimate, but it contains no proof.

\forall N \geq 4, N even \Rightarrow \exists p, q \in P : N = p + q

Known results and sources

Goldbach asks whether every even integer can be reached by a pair of primes; the source's analytic route remains incomplete.

Research problem

Exact mathematical statement

Every even integer $N\ge 4$ is a sum of two primes. Equivalently, for every such $N$ , there exist primes $p$ and $q$ with

N=p+q.

The source uses a smooth symmetric cutoff to define a weighted count $G_\eta(N)$ ; positivity of that count gives an actual Goldbach representation.

Problem infographic

Problem at a glance

A wide diagram moves from prime-pair arcs over even tokens to a weighted positive count and an analytic region whose narrow rust collar remains unresolved. — The source reduces prime-pair existence to positivity and separates candidate-controlled ranges from a still-open multiplier collar.

Current mathematical picture

Where work on Goldbach's Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureWithdrawn Kloosterman-fraction improvement

The source records that a missing factor changes an L^5 contribution to L^7 and labels the resulting 11/23 exponent invalid. The published Bettin-Chandee theorem remains an established input after exact hypothesis matching, and the symmetric collar route remains open.

Route status · Narrowed route

Main reductionExact lower bound

The source defines an exact lower bound L_N no larger than the weighted Goldbach count.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Goldbach's Conjecture in numbers

4.2kretained lines of mathematical investigation4,188 in the current working snapshot

Argument development: 3,740 · 89%
Explored or eliminated routes: 124 · 3%
Computational analysis: 28 · 1%
Open obligations: 104 · 2%
Definitions and setup: 192 · 5%

6selected mapped statements2routes investigated5open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byChristian GoldbachProposed1742Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.

Suggested move: Derive a smooth symmetric partition with explicit overlap corrections and retain the exact H-to-C operator shift.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeWithdrawn Kloosterman-fraction improvement

Route status · Narrowed route

Narrowed routeUnshifted one-sided collar estimation

The failure-certificate list rejects both the missing minus-two shift and independent estimates of isolated slices. Use the H-normalized operator or the shifted numerator operator on an exact symmetric grouping with controlled principal parts.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

Prove the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.Suggested move: Derive a smooth symmetric partition with explicit overlap corrections and retain the exact H-to-C operator shift.

Ready to work on

Upgrade every candidate-closed non-collar block to a publication-grade theorem with uniform parameter control.Suggested move: Write the dyadic and Mellin decomposition line by line, including small-height transitions and all gcd sums.

Ready to work on

Symmetric collar estimate

The shifted recombined symmetric collar estimate is the central new theorem still missing from this route.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Non-collar audit

Dyadic, Mellin, transition, gcd, boundary, and complex-parameter checks are required before the candidate region becomes a theorem.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

After asymptotic closure, provide effective constants and certified finite verification below the threshold.Suggested move: Extract explicit error constants, isolate the threshold, and independently verify every remaining even integer in the finite range.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

Binary Goldbach remains open. It has been verified through 4×10^18, while the related ternary conjecture is proved; neither result supplies a universal binary proof.

[2][4][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2014
Computational resultOliveira e Silva, Herzog, and Pardi reported verification of binary Goldbach for every even integer through 4×10^18.[2]
2013
Peer reviewedHelfgott proved the ternary Goldbach conjecture, the distinct assertion that every odd integer greater than five is a sum of three primes.[3]
1742
Historical sourceThe modern binary and ternary Goldbach problems trace to the Goldbach–Euler correspondence.[1][3]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

3 standing statements3 proposed statements5 open questions2 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction3 of 63
lemma1 of 61
equivalence1 of 61

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included

retained route statementEach even N at least 4 equals p plus q for primes p and q.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementPositive weighted countintermediate
retained route statementExact lower boundintermediate
retained route statementCandidate-closed non-collar regionintermediate
retained route statementWithdrawn exponent excludedintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureWithdrawn Kloosterman-fraction improvementreported failure
Useful failureUnshifted one-sided collar estimationreported failure
Research targetProve the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.open
Research targetUpgrade every candidate-closed non-collar block to a publication-grade theorem with uniform parameter control.open
Research targetAfter asymptotic closure, provide effective constants and certified finite verification below the threshold.open
Research targetSymmetric collar estimateopen
Research targetNon-collar auditopen
ComputationThe current work reports symbolic and finite assembly checks of shift identities, endpoint derivatives, a small zero mode, contour constraints, and source-status warnings.These checks reduce transcription risk but the source explicitly states that they do not replace the remaining analytic proofs. · reported unreproduced
Narrowed routeWithdrawn Kloosterman-fraction improvementThe source records that a missing factor changes an L^5 contribution to L^7 and labels the resulting 11/23 exponent invalid. The published Bettin-Chandee theorem remains an established input after exact hypothesis matching, and the symmetric collar route remains open.
Narrowed routeUnshifted one-sided collar estimationThe failure-certificate list rejects both the missing minus-two shift and independent estimates of isolated slices. Use the H-normalized operator or the shifted numerator operator on an exact symmetric grouping with controlled principal parts.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.

Goldbach's Conjecture · ready to start

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Research contextPrepared context for any AI agent

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Current routes and known obstacles
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
The Euler Archive: Christian Goldbachauthoritative webpage · Mathematical Association of America · accessed Aug 14, 2026
2
Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4×10^18peer reviewed result · Tomás Oliveira e Silva, Siegfried Herzog, Silvio Pardi · Mathematics of Computation · 2014 · DOI 10.1090/S0025-5718-2013-02787-1 · MR MR3194140 · accessed Aug 14, 2026
3
The ternary Goldbach conjecture is truepeer reviewed result · Harald Andrés Helfgott · arXiv and Annals of Mathematics Studies · 2013-12-30 · ARXIV 1312.7748 · accessed Aug 14, 2026
4
Hempel's Ravens and the Goldbach Conjectureauthoritative webpage · Marc Lange · American Mathematical Society · 2023 · accessed Aug 14, 2026

Important qualifications

Scoped primary and official status pass, not a systematic survey of all analytic progress.
Finite verification and the solved ternary theorem do not establish binary Goldbach.
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Goldbach's Conjecture

Exact mathematical statement

Problem at a glance

Where work on Goldbach's Conjecture stands

Goldbach's Conjecture in numbers

Prove the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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