This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple ( x, y, z ) ∈ Z 3 such that x 2 + y 2 = z 2 . The name comes from elementary geometry: if a right triangle has leg lengths x and y and hypotenuse length z , then x 2 + y 2 = z 2 . Of course here x, y, z are posi- tive real numbers. For most integer values of x and y , the integer x 2 + y 2 will not be a perfect square, so the positive real number √ x 2 + y 2 will be irrational: e.g. x = y = 1 = ⇒ z = √ 2. However, a few integer solutions to x 2 + y 2 = z 2 are familiar from high school algebra (and the SATs): e.g. (3 , 4 , 5), (5 , 12 , 13). Remark: As soon as we have one solution, like (3 , 4 , 5), we can find infinitely many more, however in a somewhat unsatisfying way. Namely, if ( x, y, z ) is a Pythagorean triple and a is any integer, then also ( ax, ay, az ) is a Pythagorean triple: ( ax ) 2 + ( ay ) 2 = a 2 ( x 2 + y 2 ) = a 2 z 2 = ( az ) 2 . This property of invariance under scaling is a characteristic feature of solutions ( x 1 , . . . , x n ) to homogeneous polynomials P ( t 1 , . . . , t n ) in n-variables. We recall what this means: a monomial is an expression of the form ct a 1 1 ··· t a n n (for a nonzero constant c ), and the degree of the monomial is defined to be a 1 + . . . + a n , i.e., the sum of the exponents. A polynomial is said to be homogeneous of degree d if each of its monomial terms has degree d , and simply homogeneous if it is homogeneous of some degree d . For instance, the polynomial P ( x, y, z ) = x 2 + y 2 − z 2 is homogeneous of degree 2, and indeed for any N the Fermat polynomial P N ( x, y, z ) = x N + y N − z N is homogeneous of degree N . Moreover, every (nonconstant) homogeneous polyno- mial P ( t 1 , . . . , t n ) has zero constant term, hence P (0 , . . . , 0) = 0. So (0 , . . . , 0) is a solution to any homogeneous polynomial, called the trivial solution . Coming back to Pythagorean triples, these considerations show that for all a ∈ Z , (3 a, 4 a, 5 a ) is a Pythagorean triple (again, familiar to anyone who has studied for the SATs). For many purposes it is convenient to regard these rescaled solutions as being equivalent to each other. To this end we define a Pythagorean triple ( a, b, c ) to be primitive if gcd( a, b, c ) = 1. Then every nontrivial triple ( a, b, c ) is a positive integer multiple of a unique primitive triple, namely ( a d , b d , c d ) where d = gcd( a, b, c ). Thanks to Katelyn Andrews, Laura Nunley and Kelly Payne for pointing out typos. 1 2 PETE L. CLARK Our goal is to find all primitive Pythagorean triples. There are many ways to do so. We prefer the following method, both for its simplicity and because it moti- vates the study of not just integral but rational solutions of polynomial equations....
View Full Document