Chapter 14.3, Problem 5MC

Mathematical Connections

A Pythagorean triple is a sequence of three natural numbers $a$, $b$ and $c$ that satisfy the relationship $a^2+b^2=c^2$. The least three numbers that form a Pythagorean triple are $3-4-5$. Another triple is $5-12-13$ because $5^2+{12}^2={13}^2$.

a. Find two other Pythagorean triples.

b. Does doubling each number in a Pythagorean triple result in a new Pythagorean triple? Why or why not?

c. Does adding a fixed number to each number in a Pythagorean triple result in a new Pythagorean triple? Why or why not?

d. Suppose $a=2uv$, $b=u^2-v^2,$ and $c=u^2+v^2$, where $u$ and $v$ are natural numbers with $u>v$. Determine whether $a-b-c$ is a Pythagorean triple.