What Are Some Common Pythagorean Triples?

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A Pythagorean triple is a set of three positive integers, (a, b, c), such that a right triangle can be formed with the legs a and b and the hypotenuse c. The most common Pythagorean triples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25).

By the Pythagorean theorem, all Pythagorean triples must obey the equation a² + b² = c², which is geometrically represented as a right triangle with the shorter legs a and b and the longer hypotenuse c. For any Pythagorean triple, the product of the legs is always divisible by 12 and the product of all three sides is divisible by 60. There has been no proof excluding the possibility that two distinct triples may have the same product. Specific Diophantine equations are solved to generate Pythagorean triples.