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History, Definition, and Calculations of Pythagorean Theorem

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1) The numbers 3, 4, and 5 are called Pythagorean triples since 3 2+ 4 2= 5 2. The numbers 5, 12, and 13 are also Pythagorean triples since 5 2 + 12 2= 13 2. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples.

2) Select at least 5 more Pythagorean Triples. Show why your 5 sets of Pythagorean triples work in the Pythagorean Formula. They should all be different from each other, and should not be the multiple of another. All 5 different Pythagorean Triples should be done using the rule that allows you to find ALL the Pythagorean triples, show in detail using the rule how they were created.

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Solution Summary

In this solution I describe the series of formulas which can be used to generate Pythagorean triples. I explain these formulas in their historical context. I then go on to demonstrate how to derive any Pythagorean triple, step by step.

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1) Pythagoras himself created a formula for generating triples (circa 540 B.C.) which is:
m2 + ((m2 - 1)/2)2 = ((m2 + 1)/2)2, where m must be odd.

Plato is credited with another formula (circa 380 B.C.) which is:
(2m)2 + (m2 - 1)2 = (m2 + 1)2, where m is any natural number.

Euclid (circa 300 B.C.) presented the most general formula for obtaining Pythagorean triples in his book Elements. This formula is:
(2mn)2 + (m2 - n2)2 = (m2 + n2)2, where u and v have no common ...

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