Break through!
Let's just look at the solutions when is positive.
Let be all ordered solutions i.e.
I claim the solutions for paired with is . (Can you see why?)
This is big because now we can find all solutions quite easily!
. Let's see if you can find the formula for now.
I count from zero!
I haven't tried yet; I've got too many things to do tonight! No good idea comes to mind though. I'll give it a shot tomorrow if I can.
Now I guess all that's left is to show the odd-labeled Fibonacci numbers satisfy .
I'm having a lot of trouble showing this though... what's your take on this?
One more thing: you need to show this is the only sequence that generates positive solutions. I've figured out a way, but let's see you try first.
We want to show .
Now .
Therefore it suffices to show . (Can you see why?)
So... .
This last result can be shown inductively (I omit the details, but have verified it).
Since we know this is the only sequence of positive integers that satisfies this equation, we have all solutions to with :
. (When , .)
The solutions with follow analogously.
I already did use the excel to show the answers for that equation. Since I received this Equation as an assignment, I could really use some help. My teacher pointed me to the direction of assigning Fibonnacci numbers as algebraic expressions. Sorry for the late responses I usually don't have access to a computer regularly.