Question

8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean triple where a is the even number: Then and are perfect squares, say s2 and t2, espectively; and s and t are relatively prime. So now we can completcly characterize all primitive Pythagorean triples 8.7 Theorem (Pythagorean Triple Theorem). Let (a, h, c) be a triple of natural mmbers with a even, b odd, and c odd. Then (a. b. c) is aprimitive Pythagorean triple if and only if there exist relatively prime pasitive integers s and t, one even and one odd, such that a 2st, b (2-2). and c(2+ The formulas given in the Pythagorcan Triple Theorem allow us to in- vestigate the types of numbers that can occur in Pythagorcan triples. ILct's start our investigation by looking at examples 8.8 Exercise. Using the above formulas make a lengthy list of primitive Pythagorean triples We'll begin by looking at the legs and then think about the hypotenuse later 8.9 Exercise. Make a conjecture that describes those natural numbers that can appear as legs in a primitive Pythagorean triple.

Answer 1

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