We say (a, b, c) is a Pythagorean Triple if a, b, c are positive integers and a? 3² = c². For example, (3, 4, 5) and (5, 12, 13) are Pythagorean triples. Assume that (a, b, c) is a Pythagorean triple in which the only common divisors of a, b, c are ±1. 1. Show that a and b cannot both be odd. 2. Assume that a is even. Show that there exist relatively prime integers m and n so that а %3 2тп, b %3D т? — п?, аnd c %3 т? + п?. Hint: Factor a² = c² – b² after showing that (c+b, c – b) = 2.
We say (a, b, c) is a Pythagorean Triple if a, b, c are positive integers and a? 3² = c². For example, (3, 4, 5) and (5, 12, 13) are Pythagorean triples. Assume that (a, b, c) is a Pythagorean triple in which the only common divisors of a, b, c are ±1. 1. Show that a and b cannot both be odd. 2. Assume that a is even. Show that there exist relatively prime integers m and n so that а %3 2тп, b %3D т? — п?, аnd c %3 т? + п?. Hint: Factor a² = c² – b² after showing that (c+b, c – b) = 2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We say (a, b, c) is a Pythagorean Triple if a, b, c are positive integers and a? +
6² = c². For example, (3, 4, 5) and (5, 12, 13) are Pythagorean triples. Assume that (a, b, c) is a
Pythagorean triple in which the only common divisors of a, b, c are ±1.
||
1. Show that a and b cannot both be odd.
2. Assume that a is even. Show that there exist relatively prime integers m and n so that
а %3D 2тп, b — т? — п2, and c — т? + п?.
Hint: Factor a² = c² – b² after showing that (c+b, c – b) = 2.
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