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Clay · Millennium Problems

Clay Mathematics Institute

Open source ↗

Overview

Seven Millennium Prize Problems; five selected here

4 cited passages ↓5 problems ↓Record observed-2026-10-10 ↓

Passages

The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.

Passage index · 4
  1. https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/
  2. https://www.claymath.org/millennium/hodge-conjecture/
  3. Clay Hodge page, overview: dimension less than four
  4. https://www.claymath.org/millennium/riemann-hypothesis/
Passage 1 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s = 1?

Read at source ↗2 recorded uses ↓

Replaces an earlier passage. View versioned records →

Passage details

Passage ID: birch-and-swinnerton-dyer-conjecture:entry:0:v2

Source item: https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

Source record: clay-millennium@observed-2026-10-10

Replaces passage: birch-and-swinnerton-dyer-conjecture:entry:0:v1

Passage 2 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/hodge-conjecture/

On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?

Read at source ↗2 recorded uses ↓

Replaces an earlier passage. View versioned records →

Passage details

Passage ID: hodge-conjecture:entry:0:v2

Source item: https://www.claymath.org/millennium/hodge-conjecture/

Source record: clay-millennium@observed-2026-10-10

Replaces passage: hodge-conjecture:entry:0:v1

Passage 3 · OpenMath paraphrasePermalink

Clay Hodge page, overview: dimension less than four

The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.

Read at source ↗3 recorded uses ↓
Passage details

Passage ID: hodge-conjecture:hodge-dimension-at-most-three:v2

Source item: Clay Hodge page, overview: dimension less than four

Source record: clay-millennium@observed-2026-10-10

Passage 4 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/riemann-hypothesis/

Do all nontrivial zeros of the Riemann zeta function have real part 1/2?

Read at source ↗2 recorded uses ↓

Replaces an earlier passage. View versioned records →

Passage details

Passage ID: riemann-hypothesis:entry:0:v2

Source item: https://www.claymath.org/millennium/riemann-hypothesis/

Source record: clay-millennium@observed-2026-10-10

Replaces passage: riemann-hypothesis:entry:0:v1

Also cited · observed-2026-10-01

Clay Mathematics Institute

Seven Millennium Prize Problems; five selected here

Passage 1 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s = 1?

Read at source ↗2 recorded uses ↓
Passage details

Passage ID: birch-and-swinnerton-dyer-conjecture:entry:0:v1

Source item: https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

Source record: clay-millennium@observed-2026-10-01

Passage 2 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/hodge-conjecture/

On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?

Read at source ↗2 recorded uses ↓
Passage details

Passage ID: hodge-conjecture:entry:0:v1

Source item: https://www.claymath.org/millennium/hodge-conjecture/

Source record: clay-millennium@observed-2026-10-01

Passage 3 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/p-vs-np/

Is every decision problem with solutions verifiable in polynomial time also solvable in polynomial time?

Read at source ↗2 recorded uses ↓
Passage details

Passage ID: p-vs-np:entry:0:v1

Source item: https://www.claymath.org/millennium/p-vs-np/

Source record: clay-millennium@observed-2026-10-01

Passage 4 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/poincare-conjecture/

Is every closed, simply connected three-dimensional manifold homeomorphic to the three-sphere?

Read at source ↗4 recorded uses ↓
Passage details

Passage ID: poincare-conjecture:entry:0:v1

Source item: https://www.claymath.org/millennium/poincare-conjecture/

Source record: clay-millennium@observed-2026-10-01

Passage 5 · OpenMath paraphrasePermalink

https://www.claymath.org/millennium/riemann-hypothesis/

Do all nontrivial zeros of the Riemann zeta function have real part 1/2?

Read at source ↗2 recorded uses ↓
Passage details

Passage ID: riemann-hypothesis:entry:0:v1

Source item: https://www.claymath.org/millennium/riemann-hypothesis/

Source record: clay-millennium@observed-2026-10-01

Clay Mathematics Institute

Record details
Source ID
clay-millennium
Record ID
clay-millennium@observed-2026-10-01
Recorded title
Clay · Millennium Problems
Revision
observed-2026-10-01
Observed
Oct 1, 2026
Source updated
Not recorded
Content SHA-256
Not recorded
Update method
Manual checked import; new observations require staff preview and publication
Identifier policy
Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge
Check interval
30 days
Rights
metadata-link
License
Not established; links and OpenMath paraphrases only

Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.

Source terms ↗

Used in

Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.

Birch and Swinnerton-Dyer conjecture →

Problem publication v17

Problem context · 4 citations
Earlier source assessment

Source assessment · superseded assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 1, 2026

Assessment source ↗View in problem →

Question

Birch and Swinnerton-Dyer conjecture

View in problem →

Source assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 10, 2026

Assessment source ↗View in problem →

Hodge conjecture →

Problem publication v15

All codimensions in dimension ≤ 3 →

Source assessment

Proved

All codimensions in dimension ≤ 3

Every codimension, restricted to complex dimension ≤ 3.

Observed Oct 10, 2026

Assessment source ↗Statement v3 →

Statement

Statement v3 →
Problem context · 5 citations
Earlier source assessment

Source assessment · superseded assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 1, 2026

Assessment source ↗View in problem →

Question

Hodge conjecture

View in problem →

Source assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 10, 2026

Assessment source ↗View in problem →

Poincaré conjecture →

Problem publication v13

The topological three-dimensional Poincaré theorem →

Source assessment

Proved

Solved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 1, 2026

Assessment source ↗Statement v2 →

Scope

The topological three-dimensional Poincaré theorem

Statement v2 →

Statement

Statement v2 →
Problem context · 1 citations

Question

Poincaré conjecture

View in problem →

Riemann hypothesis →

Problem publication v13

Problem context · 4 citations
Earlier source assessment

Source assessment · superseded assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 1, 2026

Assessment source ↗View in problem →

Question

Riemann hypothesis

View in problem →

Source assessment

Open

Unsolved

The cited source item and version only; this is not an independent OpenMath assessment.

Observed Oct 10, 2026

Assessment source ↗View in problem →

Credits

Clay Mathematics Institute

Versions

OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.

  1. observed-2026-10-10 · selectedObserved Oct 10, 2026 · used in current publications

    Clay · Millennium Problems

  2. observed-2026-10-01Observed Oct 1, 2026 · used in current publications

    Clay · Millennium Problems

Record details
Source ID
clay-millennium
Record ID
clay-millennium@observed-2026-10-10
Recorded title
Clay · Millennium Problems
Revision
observed-2026-10-10
Observed
Oct 10, 2026
Source updated
Not recorded
Content SHA-256
Not recorded
Update method
Manual checked import; new observations require staff preview and publication
Identifier policy
Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge
Check interval
30 days
Rights
metadata-link
License
Not established; links and OpenMath paraphrases only

Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.

Source terms ↗