Math 250: Pythagorean Triples We all are familiar with the Pythagorean Theorem, which states that the lengths a and b of the legs Summary: Math 250: Pythagorean Triples We all are familiar with the Pythagorean Theorem, which states that the lengths a and b of the legs of a right triangle, together with the length c of the hypotenuse, satisfy the relation a2 + b2 = c2 . The triple (a, b, c) is a Pythagorean Triple iff a, b, and c satisfy the Pythagorean Theorem. A Pythagorean triple is primitive iff a, b, and c have no common integer divisor except 1. A pair of numbers p and q is relatively prime iff they have no common integer divisor except 1. Note that the converse of the Pythagorean Theorem is also true. Throughout the discussion below, we need to specify the parity of an integer n, namely whether it is even (n = 2j for some integer j) or odd (n = 2k + 1 for some integer k). We will let N be the set of natural numbers {1, 2, 3, · · · }, and Z the set of integers {0, ±1, ±2, · · · }. 1. Prove that p is odd iff p2 is odd, and q2 is even iff q is even. Hint: To show p2 odd = p odd, consider p2 - 1. 2. If n Z, then n2 Collections: Mathematics