Pythagorean triples
E530315
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, representing the side lengths of right-angled triangles.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| primitive Pythagorean triples | 0 |
| non-primitive Pythagorean triples | 0 |
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
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.