As we have just seen, we get every Pythagorean triple (a, b, c) with b even from the formula
(a, b, c) = (u2 − v2, 2uv, u2 + v2)
by substituting in different integers for u and v. For example, (u, v) = (2, 1) gives the smallest triple (3, 4, 5).
(a) If u and v have a common factor, explain why (a, b, c) will not be a primitive Pythagorean triple.
(b) Find an example of integers u > v > 0 that do not have a common factor, yet the Pythagorean triple (u2 − v2, 2uv, u2 + v2) is not primitive. .
(c) Make a table of the Pythagorean triples that arise when you substitute in all values of u and v with 1 ≤ v ≤ 10. .
(d) Using your table from (c), find some simple conditions on u and v that ensure that the Pythagorean triple (u2 − v2, 2uv, u2 + v2) is primitive. .
(e) Prove that your conditions in (d) really work.