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Wikipedia
https://en.wikipedia.org/wiki/Pythagorean_triple
Pythagorean triple - Wikipedia
WebA Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is …
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May 18, 2024 - YouTube
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Pythagorean triple
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A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written, a well-known example is. If is a Pythagorean triple, then so is for any positive …
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A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written, a well-known example is. If is a Pythagorean triple, then so is for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which a, b and c are coprime. For example, is a primitive Pythagorean triple whereas is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a²+b²=c²; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=√(2) is a right triangle, but (1,1,√(2)) is not a Pythagorean triple because √(2) is not an integer. Moreover, 1 and √(2) do not have an integer common multiple because √(2) is irrational. Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10. When searching for integer solutions, the equation a² + b² = c² is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.
Wikipedia
When you search something for integer solution then this is a Diophantine equation.
andlearning.org
Pythagorean triples are into existence since the ancient times from 1800 BC or more. However, the concept became clearer after 1900 when it was used practically.
For example, a Pythagorean triplet (3, 4, 5) is given to us, if we multiply the triples with 3, it will give (9, 12, 15). 1.
geeksforgeeks.org
The scaling of the pythagoras triples is multiplying the given triples with some natural numbers and observing that the new set generated satisfies the Pythagorean triples condition too.
t t t t t t t t t t t t ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ String pulled taut t t t t t t t t t t t t
math.brown.edu
Even more amazing is the fact that the Babylonians created tables of quite large Pythagorean triples, which they may have used as primitive trigonometric tables.
Explore more
Pythagorean theorem
Congruent number
Pythagoreanism
Natural number
Trigonometric functions
Image: Getty Images. Data: Wikipedia · andlearning.org · geeksforgeeks.org · math.brown.edu
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