Choose positive integers m and n, with m

Question

Choose positive integers m and n, with m < n. Let x = 2mn, y = n^2 −m^2, and z = m^2 + n^2. It so happens that these three positive integers x, y, and z have a special property. What is the property? Can you prove a general result?

calculusxy · Answer

@agent0smith

calculusxy · Answer

@skullpatrol @Nnesha @Directrix

calculusxy · Answer

I had n = 4 and m = 2. So I got: 
x = 16
y = 12
z = 20

calculusxy · Answer

I don't know what's so special about these three numbers though except they are the multiplies of 2 and 4?

directrix · Answer

Do you know what is special about this triple of numbers:  ( 3, 4, 5 )  ?

calculusxy · Answer

They are Pythagorean Triples! That makes so much more sense!

calculusxy · Answer

16/4 = 4
12/4 = 3
20/4 = 5

calculusxy · Answer

What do they mean by proving the "general result" ?

directrix · Answer

Correct. Now, your task is to prove a general result for that with the variables that were given.

calculusxy · Answer

I still don't understand.

agent0smith · Answer

x = 2mn, y = n^2 −m^2, and z = m^2 + n^2

Can't you just show that  $\large x^2 + y^2 = z^2 $ ?

agent0smith · Answer

This looks like it'd be true to me, from some mental checking$$\large (2mn)^2 + (n^2 - m^2)^2 = (m^2 + n^2)^2$$