congruent number
Let an agent explain congruent numbers and the congruent number problem, relay known results, Tunnell theorem and the link to elliptic curves from mathematical references, and distinguish congruent numbers from the unrelated notion of congruence.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a congruent number is.
Definition
Definition.
Definition
Definition.
- What is a congruent number, and why is 5 congruent but 1, 2 and 3 not? definition
- Is the question about congruent numbers or about congruence modulo n? boundary
Examples
Examples.
Examples
Examples.
- What are the small congruent numbers, and what rational triangles realise them? measurement
- Which entry fits rational right triangle? action
Theory Theory.
Elliptic
Elliptic curves.
Elliptic
Elliptic.
- How does n being congruent correspond to the elliptic curve y^2 = x^3 - n^2 x having positive rank? provenance
- Which references are standard? provenance
Tunnell
Tunnell theorem.
Tunnell
Tunnell.
- What does Tunnell theorem say, and why does it depend on the Birch and Swinnerton-Dyer conjecture? provenance
- Which entry fits Birch and Swinnerton-Dyer conjecture? action
Problem The open problem.
Status
Status.
Status
Status.
- What is known and unknown about classifying congruent numbers? provenance
- Which sources are cited? provenance
Computation
Computation.
Computation
Computation.
- How have large computations enumerated congruent numbers? provenance
- Which entry fits computational number theory? action
Context History.
History
History.
History
History.
- How did the problem arise in Arabic manuscripts and Fibonacci work, and how did Fermat show 1 is not congruent? provenance
- Which entry fits the history of number theory? action
Culture
Popular mathematics.
Culture
Culture.
- How is the congruent number problem presented in popular mathematics? provenance
- Which entry fits unsolved problems in mathematics? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A congruent number is a positive integer that is the area of a right triangle with rational side lengths, such as 5, 6, 7, 20, 21 and 24. The congruent number problem has roots in ancient Arabic and medieval mathematics and remains open in general. A positive integer n is congruent if and only if the elliptic curve y^2 = x^3 - n^2 x has positive rank over the rational numbers, an unconditional equivalence. For squarefree positive integers, Tunnell's theorem gives an unconditional necessary criterion for congruence; sufficiency of that criterion depends on the Birch and Swinnerton-Dyer conjecture.
Why it exists Filled
Let an agent explain congruent numbers and the congruent number problem, relay known results, Tunnell theorem and the link to elliptic curves from mathematical references, and distinguish congruent numbers from the unrelated notion of congruence.
Distinguishing features Filled
- Rational right triangle area
- Ancient origin
- Elliptic curve connection
- Open problem
What robots and AI may and may not do Filled
Must not
- Present a result that depends on an unproved conjecture as unconditional.
- Claim a number is or is not congruent without a check or a source.
- Misattribute historical or modern results to the wrong mathematicians.
Only with a human decision
- Announcing a claimed proof of an open problem in this area.
May
- Test, list and explain congruent numbers with stated methods and sources.
- Make clear which results depend on unproved conjectures.
Moral aspects Filled
- Credit for results in number theory belongs to those who proved them.
- False claims about open problems mislead students and waste reviewers' time.
Who is affected
- Mathematicians
- Students
Owners Filled
Steward
Nobody owns the concept; it is held in common.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- Tunnell theorem - conditionally
- elliptic curve - y^2 = x^3 - n^2 x
- OEIS - as A003273
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.congruent-number
- Wikidata: Q325978 (https://www.wikidata.org/wiki/Q325978)
- OEIS: A003273 congruent numbers
Direct properties not applicable Not applicable
- smallest congruent number: 5 value - triangle 3/2, 20/3, 41/6
- congruent numbers below 100: 37 count - OEIS A003273
- Tunnell theorem: 1983 year
Plane XCT: no invented physical properties.
Recognition optional Filled
- Area of a rational right triangle
- Congruent numbers 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24
- Congruence modulo n is a different concept; Pythagorean triples give integer-sided triangles
- Not a visible object; integers and rational triangles.
Capabilities and actions required Filled
- explain the definition
- relay known results
- describe the elliptic curve link
- distinguish from congruence
Hazards and failure modes required Filled
- Confusing with modular congruence
- Overstating what is proved unconditionally
- Missing the squarefree reduction
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- congruent numbers
- non-congruent numbers such as 1, 2, 3 and 4
- squarefree congruent numbers as the essential cases
- residue class results such as numbers congruent to 5, 6 or 7 mod 8 under BSD
Sources Filled
- Wikidata item Q325978: congruent number - identity and sense of the item
- Wikipedia: Congruent number - general description of the item
Open questions
- Should the congruent number problem be a separate entry?
- How should mathematical references be linked?
- How should computational results such as the 2009 trillion-number computation be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q325978/spec.json