Pythagorean triple
Enable an AI agent to recognise an exact Pythagorean triple, assess its arithmetic properties, and determine valid transformations and right-triangle uses.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Codex + Grok
Purpose and description
Enable an AI agent to recognise an exact Pythagorean triple, assess its arithmetic properties, and determine valid transformations and right-triangle uses.
A Pythagorean triple is a triple of positive integers (a, b, c) satisfying a² + b² = c², equivalently the side lengths of a right-angled triangle with integer sides.
It can be Verify or reject a candidate using exact integer arithmetic.; Canonicalise leg order and detect equivalent representations.; Reduce a valid triple to its primitive representative or generate positive integer multiples.; Construct triples from admissible Euclidean parameters and check a proposed generation witness.; Complete two proposed side values by testing whether the required third value is a positive integer.; Compute exact perimeter and area and assess specified right-triangle constraints..
Distinguishing features
All three entries must be positive integers under the adopted convention; a right triangle with an irrational side is not a Pythagorean triple.
After identifying c as the largest entry, a² + b² must equal c² exactly; approximate equality is insufficient.
The hypotenuse c is strictly greater than either leg; zero-length and degenerate cases are excluded under the adopted convention.
A common divisor greater than one makes the triple nonprimitive, not invalid: both (3, 4, 5) and (6, 8, 10) qualify.
Exchanging the two legs preserves the underlying triple; multiplying all entries by an integer greater than one produces a distinct triple in the same primitive family.
Scope
+ Exact integer membership and verification of the Pythagorean equation
+ Leg and hypotenuse roles, leg ordering, and canonical representation
+ Greatest common divisor, primitive reduction, and integer scaling
+ Euclidean generation parameters and their validity conditions
+ Right-triangle interpretation and exact derived quantities
- General real-valued right triangles and their measurement uncertainty
- Rational points on the unit circle as independently modelled objects
- General Diophantine equations and their solution spaces
- Physical triangles, surveying observations, and construction tolerances
- Higher-dimensional integer configurations such as Euler bricks
Characteristics
- Canonical side values
- Dimensionless positive integers a < b < c, with a and b the legs Provides a consistent representation for exact checking and comparison.
- Pythagorean residual
- Integer a² + b² − c²; zero is required Separates exact membership from near misses.
- Verification state
- Unverified candidate | verified triple | rejected candidate Prevents an unchecked proposal or failed computation from being treated as a recognised triple.
- Common scale factor
- Positive integer d = gcd(a, b, c) Determines whether the triple is primitive and supplies its reduction factor.
- Primitiveness
- Primitive when d = 1 | nonprimitive when d > 1 Controls which parity and generation conditions can be applied directly.
- Primitive representative
- Canonical triple (a/d, b/d, c/d) Links scaled triples without collapsing their distinct integer values.
- Euclidean generation witness
- Integers k ≥ 1 and m > n ≥ 1, gcd(m, n) = 1, m and n of opposite parity; legs k(m² − n²), 2kmn and hypotenuse k(m² + n²) Supports constructive generation and independent verification, allowing leg exchange.
- Right-triangle quantities
- Perimeter a + b + c in abstract length units; area ab/2 in squared abstract length units Supports geometric constraints while keeping physical units separate from integer membership.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 15 findings · 29 questions.
Integer membership Establishes whether the proposed values constitute a Pythagorean triple under the adopted convention.
Integer membership and exact equality are the decisive recognition conditions.
Side domain and roles
Identifies admissible entries and assigns the hypotenuse and leg roles.
Positive integer side assignment
A candidate must contain three positive integers, with the greatest value assigned to the hypotenuse.
- Are all three supplied values exactly known positive integers? definition
- Which value is the hypotenuse, and does the supplied role assignment agree with it being the greatest value? boundary
Exact equation check
Records the arithmetic evidence that admits or rejects the candidate.
Zero Pythagorean residual
Recognition requires an exactly zero residual; rounding and arithmetic overflow cannot establish membership.
- What is the exact integer value of a² + b² − c²? measurement
- What exact calculation or symbolic argument establishes the result without rounding or overflow? provenance
- Does the available evidence justify verifying the candidate, rejecting it, or leaving it unverified? action
Primitive family Distinguishes representation equivalence from the relationship between primitive triples and their multiples.
An agent must avoid both duplicate leg permutations and the mistaken collapse of different scaled triples.
Leg exchange equivalence
Normalises interchangeable leg positions while retaining any application-specific labels.
Canonical leg order
Sorting the legs gives a stable identity for the underlying triple; an application may separately distinguish horizontal and vertical legs.
- What is the canonical representation with the smaller leg first and the hypotenuse last? definition
- Do two records differ only by leg exchange, or do their integer values differ? boundary
Common factor reduction
Determines primitiveness and identifies the primitive family.
Primitive core and scale
Dividing by the greatest common divisor yields a primitive triple; the divisor records the original triple's integer scale.
- What are d = gcd(a, b, c) and the canonical reduced triple (a/d, b/d, c/d)? measurement
- Does a comparison target have the same primitive representative, and what scale factor relates their side values? boundary
- Would the proposed rescaling preserve positive integer entries as well as the Pythagorean equation? action
Arithmetic construction Supports parameter-based generation and arithmetic consistency checks.
A triple can be constructively certified and extended into a family without relying on a list of examples.
Euclidean parameters
Relates the triple to the standard positive integer parameterisation.
Admissible generation witness
With coprime opposite-parity m > n ≥ 1, Euclid's formulas produce a primitive triple; multiplication by k ≥ 1 gives its positive integer multiples.
- Which k, m and n reproduce the two legs and hypotenuse exactly, allowing the legs to exchange positions? measurement
- Do m and n satisfy positivity, ordering, coprimality and opposite-parity conditions? boundary
- Was this witness supplied by a source, recovered from the triple, or used to generate it? provenance
Primitive parity checks
Applies necessary arithmetic properties to the primitive representative.
Primitive side consistency
In a primitive triple, exactly one leg is even, the hypotenuse is odd, and the sides are pairwise coprime; these checks supplement exact equation verification.
- Does the primitive representative have exactly one even leg and an odd hypotenuse? measurement
- Are the primitive representative's three pairwise greatest common divisors all one? measurement
- If a necessary check fails, is the primitive classification wrong or is the candidate itself invalid? boundary
Right-triangle use Connects verified integer triples to exact geometric quantities and constrained completion tasks.
Recognition should support concrete decisions about side completion, size constraints and geometric interpretation.
Exact geometric quantities
Derives right-triangle quantities from the verified side roles.
Area, perimeter and scale
The associated right triangle has perimeter a + b + c and area ab/2; applying a physical length scale requires a separate application interpretation.
- What are the exact perimeter and area of the associated right triangle? measurement
- Are these abstract side values or integer counts of an explicitly stated common length unit? boundary
- Does the triple meet the application's specified perimeter, area or side-length bounds? action
Integer side completion
Determines whether two proposed side values can be completed to an admissible triple.
Perfect-square completion
Two legs require a positive integer square root of their squared sum; a hypotenuse and leg require a positive integer square root of their squared difference.
- Are the known values both legs, or is one explicitly the hypotenuse? definition
- Is the required squared sum or positive squared difference a perfect square? measurement
- Which completions remain after enforcing side roles, positivity and any primitiveness requirement? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Kinds and varieties
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- primitive Pythagorean triple (gcd(a, b, c) = 1; equivalently pairwise coprime, with legs of opposite parity)
- non-primitive (imprimitive) triple: a positive-integer multiple k(a, b, c) of a primitive triple, k ≥ 2
- Euclid-generated triple a = k(m² − n²), b = k(2mn), c = k(m² + n²) for integers m > n ≥ 1 and k ≥ 1 (this parametrizes all triples, up to swapping the legs)
- primitive Euclid-generated triple: the same with k = 1, gcd(m, n) = 1, and m, n of opposite parity
- ordered triple versus unordered pair of legs: (3, 4, 5) distinguished from (4, 3, 5) when orientation or which leg is even is recorded
- almost-isosceles triple: legs consecutive integers, |a − b| = 1 (e.g. 20, 21, 29)
- common-leg or twin families: distinct triples that share a leg, used in catalogs and tiling arguments
- Which of these kinds and varieties hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Identifiers and schemes
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Wikidata item - Q214371 - Item for the mathematical notion Pythagorean triple (not the Pythagorean theorem).
- Wikipedia title - Pythagorean triple - Canonical English encyclopaedia lemma.
- Which of these identifiers and schemes hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Real-world use
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Laying out a right angle on site with a 3-4-5 (or scaled) rope, tape, or story pole in carpentry, masonry, and surveying.
- School and contest number theory: listing primitives, applying Euclid's formula, and proving that one leg is divisible by 3, one by 4, and the hypotenuse or a leg by 5.
- Generating all rational points on the unit circle via (a/c, b/c), hence all rational-slope right triangles.
- Integer lattice geometry: axis-aligned or grid vectors of integer length whose sum of squares is a square.
- Historical tables of integer right triangles (e.g. Plimpton 322; lists in the Indian Śulbasūtras) used as computational examples rather than as a modern product class.
- Which of these real-world use hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Typical measurements
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- legs a, b - positive integers; smallest 3 and 4; one even, one odd in every primitive triple - 1 (dimensionless integer length)
- hypotenuse c - positive integer ≥ 5; primitives have c ≡ 1 (mod 4) - 1 (dimensionless integer length)
- common factor k - 1 for a primitive triple; unbounded for general triples - 1
- Euclid parameters m > n ≥ 1 - small positive integers in tables; coprime and opposite parity when generating primitives - 1
- area ab/2 - positive integer ≥ 6 (attained at 3-4-5), always integral because one leg is even - 1 (square of the length unit)
- inradius (a + b − c)/2 - positive integer ≥ 1 - 1 (same length unit as the sides)
- Which of these typical measurements hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Failure modes and hazards
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- A 3-4-5 layout stretched or marked inaccurately yields an angle that is not 90°, so the triple is used as if it guaranteed squareness when the physical lengths are wrong.
- Floating-point tests of a² + b² = c² on measured or computed lengths both miss true triples and accept near-misses.
- Integer overflow when forming a², b², or c² in fixed-width arithmetic, especially 32-bit, silently accepts or rejects triples.
- Admitting 0 or negative entries (valid as integer solutions of a² + b² = c²) breaks the geometric reading as a triangle.
- Counting (3, 4, 5) and (4, 3, 5) twice, or omitting non-primitive multiples, depending on whether the catalog is ordered and primitive-only.
- Which of these failure modes and hazards hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Regional variation
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- English school usage sometimes says "Pythagorean triplet" rather than "triple."
- Chinese sources treat the same objects as 勾股数 (gougu numbers) under the 勾股 theorem, not under Pythagoras' name.
- Some number-theory conventions fix a odd, b even, and a < c, so the ordered triple is unique for each unordered triangle; others list both leg orders.
- Babylonian sexagesimal tables and Vedic Śulbasūtra lists record specific triples without the modern primitive/non-primitive vocabulary.
- Which of these regional variation hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Pythagorean theorem - The theorem is the real identity relating the sides of any right triangle; a Pythagorean triple is an integer solution of that equation.
- rational-sided right triangle - Sides may be rational rather than integral; clearing a common denominator yields a Pythagorean triple, and conversely every triple scales to a rational point on the unit circle.
- Heronian triangle - Integer sides and integer area, but the triangle need not be right-angled; a Pythagorean triple always gives a Heronian triangle, the converse fails (e.g. 5-5-6 or 5-5-4, 13-14-15).
- Pythagorean quadruple (and n-tuples) - Sum of three or more integer squares equals a square (a² + b² + c² = d², etc.), not two.
- Eisenstein triple - Positive integers satisfying a² − ab + b² = c², the side lengths of a 60° integer triangle, not a 90° one.
- integer-sided (scalene) triangle that is not right - The sides fail a² + b² = c² (and the two cyclic permutations); the numerical test is exact on integers.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of Pythagorean triple this model covers, and on what evidence? provenance
Sources
- Pythagorean triple - Standard definition, primitive versus non-primitive distinction, Euclid's formula, and the 3-4-5 generating examples.
- Pythagorean Triple - Generation formulae, primitive-triple conditions on (m, n), and enumerative properties used for kinds and measurements.
- Euclid's Elements, Book X (lemma preceding Proposition 29) and the integer right-triangle construction - Classical generating construction that all primitive triples arise from two integers of opposite parity.
- Pythagorean Triangles - Monograph treatment of primitive and non-primitive triples, almost-isosceles families, and shared-leg configurations.
What the second pass must settle
- Does the registry intend only positive integer triples, or also signed and zero-containing solutions requiring an explicit extension of scope?
- Does registry identity treat leg permutations as one thing, or must ordered triples be represented separately for particular tasks?
- Should rationally proportional triples be linked through primitive representatives here, or through a neighbouring model of triangle similarity?
- What evidence must consuming agents retain for verification: exact residual calculations, generation witnesses, formal proofs, or a specified combination?