Pythagorean Triples

Carly Ayers

MAT126: Survey of Mathematical Methods

Professor Colleen Radke

November 5, 2012

Pythagorean Triples

Often described as the first pure mathematician, Pythagoras of Samos was a pre-Socratic Greek philosopher, who founded, perhaps, one of the most important mathematical theorems, although, he shared no written historical documents. The Pythagorean Theorem, a relation among the sides of a right triangle, states: “The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides” (Morris, 1997). The theorem uses the equation x2 + y2 = z2, where the exponent, z, represents the length of the hypotenuse, and the exponents, a and b, represent the lengths of the other sides of a right triangle. “A Pythagorean triple is an ordered triple (x, y, z) of three positive integers such that x2 + y2 = z2. If x, y, and z are relatively prime, then the triple is called primitive” (Rowland, 2011, Theorem 1). As our assignment this week states (Bluman, 2011, p. 620, project 4), one example of a Pythagorean triple is 3, 4, and 5, because 32 + 42 = 52 which reads 9 + 16 = 25 when solved. This is the same with the numbers 5, 12, and 13, because 52 + 122 = 132 which reads 25 + 144 = 169 when solved. In this assignment, we will test one set of formulas which will generate an infinite number of Pythagorean triples, all the while showing examples of other Pythagorean triples.

One set of formulas is noted by Amar Kumar Mohapatra and Nupur Prakash, of the Guru Gobind Singh Indraprastha University of Delhi, India, in their written work, A generalized formula to determine Pythagorean triples. “Pythagoras himself has provided a formula for infinitely many triples, namely, x = 2n + 1, y = 2n2 + 2n and z = 2n2 + 2n + 1, where n is an arbitrary positive integer” (Mohapatra & Prakash, 2010). We can now test the formula using the original three numbers listed in the instructions of...