Who created the most general method of the Pythagorean Triples?

MAT126: Survey of Mathematical Methods

Instructor: Kevin Kuznia

July 23, 2012

Who created the most general method of the Pythagorean Triples?

Pythagorean Triples have been said to have been around for many thousands of years and through those years it has been developed and redeveloped to create a simpler solution to and age old problem solving equation. However the question still stands who did create the most general formula for Pythagorean Triples?

We must first understand what a Pythagorean Triple really is to decide who created the most general solution. A Pythagorean Triple is a set of positive integers, a, b and c that fits the rule: a2 + b2 = c2. One of the easiest ways to generate Pythagorean Triples is to multiply any known Pythagorean Triple by an integer. The following Pythagorean Triple’s investigates this equation.

I will use the sides of a triple as 3,4,5 and then after each one I will verify them by using the Pythagorean Theorem equation.

1) Multiply by 2 = 6,8,10

A) 6² + 8² = 10² = 100

2) Multiply by 3 = 9,12,15

A) 9² + 12² = 15² = 225

3) Multiply by 4 = 12,16,20

A) 12² + 16² = 20² = 400

I will use the sides of a triple as 5,12,13 and then after each one I will verify them by using the Pythagorean Theorem equation.

Sides of a known triple: 5,12,13

4) Multiply by 2 = 10,24,26

A) 10² + 24² = 26² = 676

5) Multiply by 3 = 15,36,39

A) 15² + 36² = 39² = 1521

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The Pythagorean Triples 6, 8, 10; 9, 12, 15; 12, 16, 20; 10, 24, 26; 15, 36, 39 are the results of taking an already known Pythagorean Triple and multiplying it by some integer. I selected this method because it was the fastest and easiest way to produce multiple Pythagorean Triples.

While there are different formulas for creating these Pythagorean Triples it is my opinion that using the formula by Euclid is the most general...