Trigonometry/For Enthusiasts/Pythagorean Triples
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Pythagorean Triples[edit]
A Pythagorean triple has three positive integers a, b, and c, such that a^{2} + b^{2} = c^{2}. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.^{[1]} Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c).
Generating Pythagorean Triples[edit]
The integers
always form a Pythagorean triple, that is
Show it works

How was it discovered?

Examples of Pythagorean Triples[edit]
Some wellknown examples are (3, 4, 5) and (5, 12, 13).
A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).
The following is a list of primitive Pythagorean triples with values less than 100: 9:(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)
Are they all generated?

Fermat's last theorem

References[edit]
 ↑ Needs a reference