Pythagorean triples
E530315
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, representing the side lengths of right-angled triangles.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Pythagorean triples canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570621 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pythagorean triples Context triple: [Fermat's theorem on sums of two squares, relatedTo, Pythagorean triples]
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A.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
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B.
Pythagoreio
Pythagoreio is a historic coastal town and popular tourist resort on the Greek island of Samos, known for its ancient harbor and archaeological sites.
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C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
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D.
Thales’ theorem
Thales’ theorem is a fundamental result in Euclidean geometry stating that any angle inscribed in a semicircle is a right angle.
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E.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pythagorean triples Target entity description: Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, representing the side lengths of right-angled triangles.
-
A.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
-
B.
Pythagoreio
Pythagoreio is a historic coastal town and popular tourist resort on the Greek island of Samos, known for its ancient harbor and archaeological sites.
-
C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
D.
Thales’ theorem
Thales’ theorem is a fundamental result in Euclidean geometry stating that any angle inscribed in a semicircle is a right angle.
-
E.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Pythagorean triples subclass
ⓘ
mathematical concept ⓘ number-theoretic object ⓘ |
| definedAs |
Pythagorean triples with gcd(a,b,c) = 1
ⓘ
Pythagorean triples with gcd(a,b,c) > 1 ⓘ sets of three positive integers (a,b,c) satisfying a^2 + b^2 = c^2 ⓘ |
| hasApplication |
integer right triangles
ⓘ
problems in recreational mathematics ⓘ |
| hasComponentRole |
a is a leg of the right triangle
ⓘ
b is a leg of the right triangle ⓘ c is the hypotenuse of the right triangle ⓘ |
| hasCondition | a^2 + b^2 = c^2 ⓘ |
| hasConstraint | a,b,c are usually taken with a ≤ b < c ⓘ |
| hasElementType | positive integers ⓘ |
| hasExample |
(11,60,61)
ⓘ
(12,35,37) ⓘ (16,63,65) ⓘ (20,21,29) ⓘ (28,45,53) ⓘ (3,4,5) ⓘ (5,12,13) ⓘ (7,24,25) ⓘ (8,15,17) ⓘ (9,40,41) ⓘ |
| hasGenerationCondition | m and n coprime and not both odd generate primitive triples ⓘ |
| hasGenerationFormula | a = m^2 - n^2, b = 2mn, c = m^2 + n^2 for integers m > n > 0 ⓘ |
| hasHistoricalAttribution |
known to ancient Babylonians
ⓘ
studied in ancient Greek mathematics ⓘ |
| hasParityPattern | primitive triples have one leg even and one leg odd ⓘ |
| hasProperty |
can be scaled by a positive integer k to form another triple (ka,kb,kc)
ⓘ
in primitive triples, exactly one of a or b is divisible by 3 ⓘ in primitive triples, exactly one of a or b is divisible by 4 ⓘ in primitive triples, exactly one of a or b is divisible by 5 ⓘ in primitive triples, hypotenuse c is odd ⓘ infinitely many exist ⓘ primitive triples are not integer multiples of smaller triples ⓘ |
| hasSubclass |
non-primitive Pythagorean triples
ⓘ
primitive Pythagorean triples ⓘ |
| hasSymmetryProperty | (a,b,c) and (b,a,c) represent the same triple geometrically ⓘ |
| relatedTo |
Diophantine equations
ⓘ
Euclidean parameterization ⓘ Pythagorean theorem ⓘ rational points on the unit circle ⓘ solutions of x^2 + y^2 = z^2 in integers ⓘ |
| represents | side lengths of right-angled triangles ⓘ |
| satisfies | Pythagorean theorem ⓘ |
| usedIn |
algebraic number theory
ⓘ
geometry ⓘ number theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pythagorean triples Description of subject: Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, representing the side lengths of right-angled triangles.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.