taxicab number
Let an agent define taxicab numbers precisely, list the known values and their representations, distinguish them from related notions such as cabtaxi numbers, and relate them to the Hardy-Ramanujan anecdote and to Diophantine problems.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define The definition.
Definition
Definition.
Definition
Definition.
- What exactly is Ta(n), and what counts as a distinct representation? definition
- Is the question about taxicab numbers, cabtaxi numbers or a generalisation? boundary
Values
Known values.
Values
Values.
- What are the known taxicab numbers and their representations? measurement
- Which values are exact and which are bounds? boundary
Compute Computation.
Verify
Verification.
Verify
Verify.
- How is a claimed representation verified, and how are candidates searched? action
- Which entry fits Diophantine equations? action
Bounds
Bounds.
Bounds
Bounds.
- How were bounds for higher taxicab numbers found, with attribution? provenance
- Which references are standard? provenance
Theory Mathematics behind it.
Cubes
Sums of two cubes.
Cubes
Cubes.
- What is known about representing numbers as sums of two cubes, and why do multiple representations exist? provenance
- Which entry fits sums of powers? action
Generalise
Generalisations.
Generalise
Generalise.
- What generalisations to other powers and more terms exist? provenance
- Should a generalisation be a separate entry? boundary
Context History and culture.
Anecdote
Hardy and Ramanujan.
Anecdote
Anecdote.
- What is the Hardy-Ramanujan anecdote, and how is it documented? provenance
- Which entry fits Ramanujan? action
Teach
Teaching.
Teach
Teaching.
- How are taxicab numbers used in teaching and popular mathematics? action
- Which misconceptions arise? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
The smallest positive integer that can be expressed as a sum of two positive cubes in n distinct ways, written Ta(n); the first taxicab numbers are 2 for n equal to 1, 1729 for n equal to 2, and 87539319 for n equal to 3, and the name recalls the anecdote of Hardy and Ramanujan about the taxi numbered 1729.
Why it exists Filled
Let an agent define taxicab numbers precisely, list the known values and their representations, distinguish them from related notions such as cabtaxi numbers, and relate them to the Hardy-Ramanujan anecdote and to Diophantine problems.
Distinguishing features Filled
- Defined by sums of two positive cubes
- Smallest with n representations
- Only a few values known exactly
- Named for the Hardy-Ramanujan anecdote
What robots and AI may and may not do Filled
Must not
- Present an upper bound as a proven taxicab number when it is not proven minimal.
- Misstate the story of 1729 or attribute it to the wrong people.
- Confuse taxicab numbers with cabtaxi numbers or with taxicab geometry.
Only with a human decision
- Correcting a published value.
May
- Explain taxicab numbers and the known values from published sources.
- Verify that a number has the stated representations as sums of cubes.
Moral aspects Filled
- Accurate records of mathematical results protect credit and trust.
- Popular anecdotes are often repeated with errors.
Who is affected
- Students
- Mathematicians
Owners Filled
Steward
Nobody owns the concept; it belongs to mathematics.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - a property of integers
- 1729 - the famous instance
- cabtaxi number - a related sequence allowing negative cubes
- On-Line Encyclopedia of Integer Sequences - sequence A011541
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.taxicab-number
- Wikidata: Q1462591 (https://www.wikidata.org/wiki/Q1462591)
- OEIS: A011541 taxicab numbers
Direct properties not applicable Not applicable
- Ta(2): 1729 integer
- Ta(3): 87539319 integer
- known values: Ta(1) to Ta(6) count - later values have upper bounds only
Plane XCT: no invented physical properties.
Recognition optional Filled
- Smallest number with n representations as a sum of two positive cubes
- Ta(2) equals 1729 equals 1 cubed plus 12 cubed equals 9 cubed plus 10 cubed
- Cabtaxi numbers allow negative cubes and are a different sequence
- Not visible; a property of integers.
Capabilities and actions required Filled
- state the definition and known values
- verify a representation as a sum of two cubes
- distinguish taxicab from cabtaxi and generalised numbers
- point to sequence references
Hazards and failure modes required Filled
- Quoting upper bounds as exact values
- Confusing taxicab and cabtaxi numbers
- Misstating the anecdote
Standards and interfaces required Filled
- No regulation; definitions follow the literature and the OEIS
Context of use required Filled
- Not for anything; a curiosity in number theory.
- taxicab numbers Ta(n)
- cabtaxi numbers allowing negative cubes
- generalised taxicab numbers for other powers and more terms
- upper bounds for unknown values
Sources Filled
- Wikidata item Q1462591: taxicab number - identity and sense of the item
- Wikipedia: Taxicab number - general description of the item
Open questions
- Which values beyond Ta(6) can be recorded as bounds?
- How should OEIS entries be linked?
- Should 1729 be a separate entry?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1462591/spec.json